Settings
| Setting | Value |
|---|---|
| alpha_pos_lower_bound | 0.3152 |
| alpha_pos_upper_bound | 0.3235 |
| exploration_depth | 11 |
| continue_from_known_states | False |
| display_uninteresting_states | True |
| display_known_states | False |
| recurse | True |
| save_as | html |
Exploration
- BLACK denotes a block in the longest path.
- BLUE denotes a hidden block.
- RED denotes an orphaned block.
- A state where all attacker blocks are blue is a typical state where the attacker has not published anything yet.
genesis
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | genesis | $0$ | $0$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $0$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $0$ |
|
Best bounds plotted together:
(A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A) | $\alpha \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(\frac{4 \alpha^{3}}{\alpha^{2} - \alpha + 1} + \frac{\alpha^{2} \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{1}{\alpha^{2} - \alpha + 1}\right)}{\alpha^{2} - \alpha + 1} + \frac{\left(1 - \alpha\right)^{2} \left(\frac{- \alpha^{2} + \alpha + 1}{\alpha^{2} - \alpha + 1} + 1.5\right)}{\alpha^{2} - \alpha + 1}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)\right)\right)$ | $\alpha \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)\right)\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1\right) - \alpha\right)\right)\right)\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $1 - \alpha$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A) | $\alpha \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(\frac{4 \alpha^{3}}{\alpha^{2} - \alpha + 1} + \frac{\alpha^{2} \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{1}{\alpha^{2} - \alpha + 1}\right)}{\alpha^{2} - \alpha + 1} + \frac{\left(1 - \alpha\right)^{2} \left(\frac{- \alpha^{2} + \alpha + 1}{\alpha^{2} - \alpha + 1} + 1.5\right)}{\alpha^{2} - \alpha + 1}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)\right)\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $1 - \alpha$ |
|
Best bounds plotted together:
(A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, H) | $\alpha \left(2 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(\frac{4 \alpha^{3}}{\alpha^{2} - \alpha + 1} + \frac{\alpha^{2} \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{1}{\alpha^{2} - \alpha + 1}\right)}{\alpha^{2} - \alpha + 1} + \frac{\left(1 - \alpha\right)^{2} \left(\frac{- \alpha^{2} + \alpha + 1}{\alpha^{2} - \alpha + 1} + 1.5\right)}{\alpha^{2} - \alpha + 1}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)\right)$ | $\alpha \left(2 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)\right)\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1\right) - \alpha\right)\right)\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, H) | $\alpha \left(2 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(\frac{4 \alpha^{3}}{\alpha^{2} - \alpha + 1} + \frac{\alpha^{2} \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{1}{\alpha^{2} - \alpha + 1}\right)}{\alpha^{2} - \alpha + 1} + \frac{\left(1 - \alpha\right)^{2} \left(\frac{- \alpha^{2} + \alpha + 1}{\alpha^{2} - \alpha + 1} + 1.5\right)}{\alpha^{2} - \alpha + 1}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, H) | $\alpha \left(2 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)\right)\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1\right) - \alpha\right)\right)\right)$ |
|
Best bounds plotted together:
(A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H) | $\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(\frac{4 \alpha^{3}}{\alpha^{2} - \alpha + 1} + \frac{\alpha^{2} \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{1}{\alpha^{2} - \alpha + 1}\right)}{\alpha^{2} - \alpha + 1} + \frac{\left(1 - \alpha\right)^{2} \left(\frac{- \alpha^{2} + \alpha + 1}{\alpha^{2} - \alpha + 1} + 1.5\right)}{\alpha^{2} - \alpha + 1}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)$ | $\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)\right)\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1\right) - \alpha\right)\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H) | $\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(\frac{4 \alpha^{3}}{\alpha^{2} - \alpha + 1} + \frac{\alpha^{2} \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{1}{\alpha^{2} - \alpha + 1}\right)}{\alpha^{2} - \alpha + 1} + \frac{\left(1 - \alpha\right)^{2} \left(\frac{- \alpha^{2} + \alpha + 1}{\alpha^{2} - \alpha + 1} + 1.5\right)}{\alpha^{2} - \alpha + 1}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H) | $\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)\right)\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1\right) - \alpha\right)\right)$ |
|
Best bounds plotted together:
(A, 2H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A) | $\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)$ | $\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)\right)\right)\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $- \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A) | $\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A) | $\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)\right)\right)\right)$ |
|
Best bounds plotted together:
(A, 2H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, H) | $\alpha \left(2 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)$ | $\alpha \left(2 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)\right)\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, H) | $\alpha \left(2 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, H) | $\alpha \left(2 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)\right)\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right)$ | $\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H) | $\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A) | $\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)$ | $\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)$ |
|
| (1, 6) | $1 - \alpha$ | (A, 2H, A, 2H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(5 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right)$ | $\alpha \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A) | $\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A) | $\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, 2A) | $\alpha \left(5 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)$ | $\alpha \left(5 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)$ |
|
| (1, 6) | $1 - \alpha$ | (A, 2H, A, 2H, A, A) | $\alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(5 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)$ | $\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)$ |
|
| (3, 3) | $3 - \alpha$ | genesis | $0$ | $0$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [7, 8] and will selfish mine on [7, 8] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [7, 8] and will selfish mine on [7, 8] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (3, 3) | $3 - \alpha$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (3, 3) | $3 - \alpha$ | genesis | $0$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, 3A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, 3A) | $\alpha \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 6\right)\right) + \left(1 - \alpha\right) \left(4 - 2 \alpha\right)$ | $\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right)$ |
|
| (1, 6) | $1 - \alpha$ | (A, 2H, A, 2H, A, 2A) | $\alpha \left(\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{1 + \frac{\frac{6 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)$ | $\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)$ |
|
| (5, 0) | $5 - \alpha$ | genesis | $0$ | $0$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [7, 8, 9] and will selfish mine on [7, 8, 9] | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [4, 7, 8, 9] and will selfish mine on [8, 9] | $2 \alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 4\right)$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (5, 0) | $5 - \alpha$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (5, 0) | $5 - \alpha$ | genesis | $0$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, 4A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, 4A) | $\alpha \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 7\right)\right) + \left(1 - \alpha\right) \left(6 - 2 \alpha\right)$ | $\alpha \left(\left(1 - \alpha\right) \left(\frac{4 \alpha}{1 - 2 \alpha} + 5\right) + 2\right) + \left(1 - \alpha\right) \left(- \alpha + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right)$ |
|
| (1, 6) | $1 - \alpha$ | (A, 2H, A, 2H, A, 3A) | $\alpha \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 5\right)\right) + \left(1 - \alpha\right) \left(- \alpha + \frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right)$ | $\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [7, 8, 9, 10] and will selfish mine on [7, 8, 9, 10] | $\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right)$ |
|
| committed [4, 7, 8, 9, 10] and will selfish mine on [8, 9, 10] | $2 \alpha + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 5\right)$ |
|
| committed [1, 4, 7, 8, 9, 10] and will selfish mine on [9, 10] | $4 \alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 6\right)$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [1, 4, 7, 8, 9, 10] and will selfish mine on [9, 10] | $4 \alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 6\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, 5A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [7, 8, 9, 10, 11] and will selfish mine on [7, 8, 9, 10, 11] | $\left(1 - \alpha\right) \left(\frac{4 \alpha}{1 - 2 \alpha} + 5\right)$ |
|
| committed [4, 7, 8, 9, 10, 11] and will selfish mine on [8, 9, 10, 11] | $2 \alpha + \left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 6\right)$ |
|
| committed [1, 4, 7, 8, 9, 10, 11] and will selfish mine on [9, 10, 11] | $4 \alpha + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 7\right)$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{4 \alpha}{1 - 2 \alpha} + 5\right) + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [1, 4, 7, 8, 9, 10, 11] and will selfish mine on [9, 10, 11] | $4 \alpha + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 7\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{4 \alpha}{1 - 2 \alpha} + 5\right) + 2$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, 4A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [7, 8, 9, 10] and will selfish mine on [8, 9, 10] | $\alpha + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 4\right)$ |
|
| committed [4, 7, 8, 9, 10] and will selfish mine on [9, 10] | $3 \alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 5\right)$ |
|
| committed [1, 4, 7, 8, 9, 10] and will selfish mine on [10] | $6 - \alpha$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [1, 4, 7, 8, 9, 10] and will selfish mine on [10] | $6 - \alpha$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 4A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [8, 9, 10, 11] and will selfish mine on [8, 9, 10, 11] | $\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right)$ |
|
| committed [4, 8, 9, 10, 11] and will selfish mine on [10, 11] | $3 \alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 5\right)$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [4, 8, 9, 10, 11] and will selfish mine on [10, 11] | $3 \alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 5\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 3A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [8, 9, 10] and will selfish mine on [9, 10] | $\alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [8, 9, 10] and will selfish mine on [9, 10] unless recovers [4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
| committed [8, 9, 10] and will selfish mine on [9, 10] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [8, 9, 10] and will selfish mine on [9, 10] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, 3A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, 3A, H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(6 - \alpha\right)$ | $\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)$ |
|
| (4, 3) | $4 - \alpha$ | genesis | $0$ | $0$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [7, 8, 9] and will selfish mine on [8, 9] | $\alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [7, 8, 9] and will selfish mine on [8, 9] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
| committed [4, 7, 8, 9] and will selfish mine on [9] | $4 - \alpha$ |
|
| committed [4, 7, 8, 9] and will selfish mine on [9] unless recovers [1] | $4 - \alpha$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [4, 7, 8, 9] and will selfish mine on [9] unless recovers [1] | $4 - \alpha$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, 3A, H) | $\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, 3A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [7, 8, 9, 11] and will selfish mine on [8, 9, 11] | $\alpha + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 4\right)$ |
|
| committed [4, 7, 8, 9, 11] and will selfish mine on [9, 11] | $3 \alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 5\right)$ |
|
| committed [1, 4, 7, 8, 9, 11] and will selfish mine on [11] | $6 - \alpha$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [1, 4, 7, 8, 9, 11] and will selfish mine on [11] | $6 - \alpha$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, 3A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 3$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 3$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 3A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, 3A) | $\alpha \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 5\right)\right) + \left(1 - \alpha\right) \left(- \alpha + \frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right)$ | $\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)$ |
|
| (4, 3) | $3 - \alpha$ | genesis | $0$ | $0$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [8, 9, 10] and will selfish mine on [8, 9, 10] | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [8, 9, 10] and will selfish mine on [8, 9, 10] unless recovers [1, 4] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{1 + \frac{\frac{6 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [8, 9, 10] and will selfish mine on [8, 9, 10] unless recovers [1, 4] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{1 + \frac{\frac{6 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, 3A) | $\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 2A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, 2A, H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)$ |
|
| (2, 7) | $2 - \alpha$ | genesis | $0$ | $0$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (2, 7) | $2 - \alpha$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, 2A, H) | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 2A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [8, 9, 11] and will selfish mine on [9, 11] | $\alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [8, 9, 11] and will selfish mine on [9, 11] unless recovers [4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
| committed [8, 9, 11] and will selfish mine on [9, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [8, 9, 11] and will selfish mine on [9, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 2A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{2 \alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{2 \alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, 2A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, 2A, H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(4 - \alpha\right)$ | $\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)$ |
|
| (2, 6) | $2 - \alpha$ | genesis | $0$ | $0$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (2, 6) | $2 - \alpha$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, 2A, H) | $\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, 2A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, 2A, H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(6 - \alpha\right)$ | $\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)$ |
|
| (1, 9) | $1 - \alpha$ | (A, 2H, A, 2H, 2A, H, A) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{2 \alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)$ |
|
| (4, 3) | $4 - \alpha$ | genesis | $0$ | $0$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [7, 8, 10] and will selfish mine on [8, 10] | $\alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [7, 8, 10] and will selfish mine on [8, 10] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
| committed [4, 7, 8, 10] and will selfish mine on [10] | $4 - \alpha$ |
|
| committed [4, 7, 8, 10] and will selfish mine on [10] unless recovers [1] | $4 - \alpha$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [4, 7, 8, 10] and will selfish mine on [10] unless recovers [1] | $4 - \alpha$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, 2A, H, A) | $\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, 2A, H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [7, 8, 10, 11] and will selfish mine on [8, 10, 11] | $\alpha + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 4\right)$ |
|
| committed [4, 7, 8, 10, 11] and will selfish mine on [10, 11] | $3 \alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 5\right)$ |
|
| committed [1, 4, 7, 8, 10, 11] and will selfish mine on [11] | $6 - \alpha$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [1, 4, 7, 8, 10, 11] and will selfish mine on [11] | $6 - \alpha$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, 2A, H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 3$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 3$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, 2A, H, A, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 3$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 3$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, 2A, H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{2 \alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{2 \alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, 2A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, 2A, 2H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{2 \alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{2 \alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{2 \alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, 2A, 2H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 3$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 3$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, 2A, 3H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{2 \alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{2 \alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, 2A) | $\alpha \left(\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{1 + \frac{\frac{6 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)$ | $\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [8, 9] and will selfish mine on [8, 9] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [8, 9] and will selfish mine on [8, 9] unless recovers [4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
| committed [8, 9] and will selfish mine on [8, 9] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(5 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [8, 9] and will selfish mine on [8, 9] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(5 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, 2A) | $\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, A, H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right)$ | $\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, A, H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, A, H) | $\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, A, H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)$ |
|
| (1, 9) | $1 - \alpha$ | (A, 2H, A, 2H, A, A, H, A) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha} + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)$ |
|
| (2, 7) | $2 - \alpha$ | genesis | $0$ | $0$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (2, 7) | $2 - \alpha$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, A, H, A) | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, A, H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [8, 10, 11] and will selfish mine on [10, 11] | $\alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [8, 10, 11] and will selfish mine on [10, 11] unless recovers [4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
| committed [8, 10, 11] and will selfish mine on [10, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [8, 10, 11] and will selfish mine on [10, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, A, H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{2 \alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{2 \alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, A, H, A, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, A, H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, A, 2H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha} + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, A, 2H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, A, 3H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right)$ | $\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, H) | $\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(4 - \alpha\right)$ | $\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)$ |
|
| (1, 8) | $1 - \alpha$ | (A, 2H, A, 2H, A, H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}\right)\right)$ | $\alpha \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)$ |
|
| (2, 6) | $2 - \alpha$ | genesis | $0$ | $0$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (2, 6) | $2 - \alpha$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, H, A) | $\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, H, 2A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(6 - \alpha\right)$ | $\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)$ |
|
| (1, 8) | $1 - \alpha$ | (A, 2H, A, 2H, A, H, A, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
| (2, 8) | $2 - 2 \alpha$ | (A, 2H, A, 2H, A, H, 2A) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)$ |
|
| (4, 3) | $4 - \alpha$ | genesis | $0$ | $0$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [7, 9, 10] and will selfish mine on [9, 10] | $\alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [7, 9, 10] and will selfish mine on [9, 10] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
| committed [4, 7, 9, 10] and will selfish mine on [10] | $4 - \alpha$ |
|
| committed [4, 7, 9, 10] and will selfish mine on [10] unless recovers [1] | $4 - \alpha$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [4, 7, 9, 10] and will selfish mine on [10] unless recovers [1] | $4 - \alpha$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, H, 2A) | $\alpha \left(\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 3\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, 3A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [7, 9, 10, 11] and will selfish mine on [9, 10, 11] | $\alpha + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 4\right)$ |
|
| committed [4, 7, 9, 10, 11] and will selfish mine on [10, 11] | $3 \alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 5\right)$ |
|
| committed [1, 4, 7, 9, 10, 11] and will selfish mine on [11] | $6 - \alpha$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [1, 4, 7, 9, 10, 11] and will selfish mine on [11] | $6 - \alpha$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 3$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, 2A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 3$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 3$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, A, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [4, 7] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [1, 4, 7] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [4, 7] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, A, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, 2A, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, 2A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, H, A, H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 3\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{2 \alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{2 \alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{2 \alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 3$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 3$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{2 \alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{2 \alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, A, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, H, A, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
| (1, 9) | $1 - \alpha$ | (A, 2H, A, 2H, A, H, 2A) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, H, A, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, H, A, H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, 2H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}\right)\right)$ | $\alpha \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 2H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, 2H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
| (1, 9) | $1 - \alpha$ | (A, 2H, A, 2H, A, 2H, A) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, 2H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 2H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [4, 7] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [1, 4, 7] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [4, 7] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 2H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 2H, A, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 2H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 3H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, 3H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 3H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 4H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, A) | $\alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(5 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)$ | $\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, A) | $\alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(5 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, A) | $\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}\right)\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}\right)$ | $\alpha \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, H, 2A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{8 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - 2 \alpha + 2\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [9, 10] and will selfish mine on [9, 10] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [9, 10] and will selfish mine on [9, 10] unless recovers [4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
| committed [9, 10] and will selfish mine on [9, 10] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [9, 10] and will selfish mine on [9, 10] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, 3A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] unless recovers [4] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{6 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{8 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{8 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, 2A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 2$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, H, A, H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 2$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, H, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, 2H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}\right)\right)$ | $\alpha \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 2H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, 2H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, 2H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 2H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 2H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 3H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 2H, A, 3H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}}$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 3H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 2H, A, 4H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}}$ |
|
Best bounds plotted together:
(A, 2H, A, 3H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(5 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right)$ | $\alpha \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H) | $\alpha \left(\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, A) | $\alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(5 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)$ | $\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)$ |
|
| (1, 7) | $1 - \alpha$ | (A, 2H, A, 3H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}\right)\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, A) | $\alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(5 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, A) | $\alpha \left(\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, 2A) | $\alpha \left(\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{1 + \frac{\frac{6 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)$ | $\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)$ |
|
| (1, 7) | $1 - \alpha$ | (A, 2H, A, 3H, A, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}\right)$ | $\alpha \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [8, 9] and will selfish mine on [8, 9] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [8, 9] and will selfish mine on [8, 9] unless recovers [4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
| committed [8, 9] and will selfish mine on [8, 9] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(5 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [8, 9] and will selfish mine on [8, 9] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(5 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, 2A) | $\alpha \left(\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, 3A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, 3A) | $\alpha \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 5\right)\right) + \left(1 - \alpha\right) \left(- \alpha + \frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right)$ | $\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)$ |
|
| (1, 7) | $1 - \alpha$ | (A, 2H, A, 3H, A, 2A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{8 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - 2 \alpha + 2\right)$ |
|
| (4, 3) | $4 - \alpha$ | genesis | $0$ | $0$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [8, 9, 10] and will selfish mine on [8, 9, 10] | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [8, 9, 10] and will selfish mine on [8, 9, 10] unless recovers [1, 4] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{1 + \frac{\frac{6 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [8, 9, 10] and will selfish mine on [8, 9, 10] unless recovers [1, 4] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{1 + \frac{\frac{6 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, 3A) | $\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, 4A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [8, 9, 10, 11] and will selfish mine on [8, 9, 10, 11] | $\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right)$ |
|
| committed [4, 8, 9, 10, 11] and will selfish mine on [10, 11] | $3 \alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 5\right)$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [4, 8, 9, 10, 11] and will selfish mine on [10, 11] | $3 \alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 5\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, 3A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [8, 9, 10] and will selfish mine on [9, 10] | $\alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [8, 9, 10] and will selfish mine on [9, 10] unless recovers [4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
| committed [8, 9, 10] and will selfish mine on [9, 10] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [8, 9, 10] and will selfish mine on [9, 10] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, 3A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] unless recovers [4] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{6 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{8 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{8 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, 2A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 2$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, 2A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, 2A, H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)$ |
|
| (2, 7) | $2 - \alpha$ | genesis | $0$ | $0$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (2, 7) | $2 - \alpha$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, 2A, H) | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, 2A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [8, 9, 11] and will selfish mine on [9, 11] | $\alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [8, 9, 11] and will selfish mine on [9, 11] unless recovers [4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
| committed [8, 9, 11] and will selfish mine on [9, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [8, 9, 11] and will selfish mine on [9, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, 2A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{2 \alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{2 \alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, A, 2A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{8 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - 2 \alpha + 2\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [9, 10] and will selfish mine on [9, 10] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [9, 10] and will selfish mine on [9, 10] unless recovers [4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
| committed [9, 10] and will selfish mine on [9, 10] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [9, 10] and will selfish mine on [9, 10] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, A, A, H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 2$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, A, H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right)$ | $\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, A, H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, A, H) | $\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, A, H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)$ |
|
| (1, 9) | $1 - \alpha$ | (A, 2H, A, 3H, A, H, A) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha} + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)$ |
|
| (2, 7) | $2 - \alpha$ | genesis | $0$ | $0$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (2, 7) | $2 - \alpha$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, A, H, A) | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [8, 10, 11] and will selfish mine on [10, 11] | $\alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [8, 10, 11] and will selfish mine on [10, 11] unless recovers [4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
| committed [8, 10, 11] and will selfish mine on [10, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [8, 10, 11] and will selfish mine on [10, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{2 \alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{2 \alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, H, A, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, A, 2H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha} + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, 2H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, 3H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, A, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}\right)$ | $\alpha \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, A, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, A, H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}\right)\right)$ | $\alpha \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}}$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, A, H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, A, H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 3H, A, 2H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}}$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, 2H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 3H, A, 3H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}}$ |
|
Best bounds plotted together:
(A, 2H, A, 4H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 4H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}\right)\right) - \alpha + 1\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best bounds plotted together:
(A, 2H, A, 4H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 4H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}\right)$ | $\alpha \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
| (1, 8) | $1 - \alpha$ | (A, 2H, A, 4H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}\right)\right)$ | $\alpha \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (1, 8) | $1 - \alpha$ | (A, 2H, A, 4H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}\right)\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 4H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 4H, 2A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{8 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - 2 \alpha + 2\right)$ |
|
| (1, 8) | $1 - \alpha$ | (A, 2H, A, 4H, A, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [9, 10] and will selfish mine on [9, 10] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [9, 10] and will selfish mine on [9, 10] unless recovers [4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
| committed [9, 10] and will selfish mine on [9, 10] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [9, 10] and will selfish mine on [9, 10] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 4H, 3A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] unless recovers [4] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{1 + \frac{\frac{6 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{8 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{8 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 4H, 2A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 2$ |
|
Best bounds plotted together:
(A, 2H, A, 4H, A, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 4H, A, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 2H, A, 4H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 4H, A, H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 2H, A, 4H, A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 2$ |
|
Best bounds plotted together:
(A, 2H, A, 4H, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best bounds plotted together:
(A, 2H, A, 4H, A, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 4H, A, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 4H, A, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 4H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 4H, A, H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}}$ |
|
Best bounds plotted together:
(A, 2H, A, 4H, A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 4H, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}}$ |
|
Best bounds plotted together:
(A, 2H, A, 5H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 5H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) - \alpha + 1\right)$ | $\alpha \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}}$ |
|
Best bounds plotted together:
(A, 2H, A, 5H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 5H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
| (1, 9) | $1 - \alpha$ | (A, 2H, A, 5H, A) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (1, 9) | $1 - \alpha$ | (A, 2H, A, 5H, A) | $- \alpha \left(1 - \alpha\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 5H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(\frac{9}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [1, 4] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best bounds plotted together:
(A, 2H, A, 5H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 2H, A, 5H, A, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 5H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}}$ |
|
Best bounds plotted together:
(A, 2H, A, 6H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 2H, A, 6H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}}$ |
|
Best bounds plotted together:
(A, 2H, A, 6H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 2H, A, 7H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}}$ |
|
Best bounds plotted together:
(A, 3H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H) | $\alpha \left(\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(\frac{4 \alpha^{3}}{\alpha^{2} - \alpha + 1} + \frac{\alpha^{2} \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{1}{\alpha^{2} - \alpha + 1}\right)}{\alpha^{2} - \alpha + 1} + \frac{\left(1 - \alpha\right)^{2} \left(\frac{- \alpha^{2} + \alpha + 1}{\alpha^{2} - \alpha + 1} + 1.5\right)}{\alpha^{2} - \alpha + 1}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)$ | $\alpha \left(\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)\right)\right)\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1\right) - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H) | $\alpha \left(\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(\frac{4 \alpha^{3}}{\alpha^{2} - \alpha + 1} + \frac{\alpha^{2} \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{1}{\alpha^{2} - \alpha + 1}\right)}{\alpha^{2} - \alpha + 1} + \frac{\left(1 - \alpha\right)^{2} \left(\frac{- \alpha^{2} + \alpha + 1}{\alpha^{2} - \alpha + 1} + 1.5\right)}{\alpha^{2} - \alpha + 1}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H) | $\alpha \left(\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)\right)\right)\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1\right) - \alpha\right)$ |
|
Best bounds plotted together:
(A, 3H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A) | $\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)$ | $\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)\right)\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A) | $\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A) | $\alpha \left(\alpha \left(4 - \alpha\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)\right)\right)$ |
|
Best bounds plotted together:
(A, 3H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, H) | $\alpha \left(2 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)$ | $\alpha \left(2 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, H) | $\alpha \left(2 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right) - \alpha\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, H) | $\alpha \left(2 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)\right)$ |
|
Best bounds plotted together:
(A, 3H, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right)$ | $\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H) | $\alpha \left(\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, A) | $\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)$ | $\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)$ |
|
| (1, 7) | $1 - \alpha$ | (A, 3H, A, 2H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}\right)\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, A) | $\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, A) | $\alpha \left(3 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)\right)$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, 2A) | $\alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(2 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}\right) + \left(1 - \alpha\right) \left(2 - 2 \alpha\right)$ | $\alpha \left(\frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1\right) + \left(1 - \alpha\right) \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right)$ |
|
| (1, 7) | $1 - \alpha$ | (A, 3H, A, 2H, A, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}\right)$ | $\alpha \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
| (3, 4) | $3 - \alpha$ | genesis | $0$ | $0$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [8, 9] and will selfish mine on [8, 9] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [8, 9] and will selfish mine on [8, 9] unless recovers [1, 5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(5 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (3, 4) | $3 - \alpha$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (3, 4) | $3 - \alpha$ | genesis | $0$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, 3A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, 3A) | $\alpha \left(2 \alpha + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 5\right)\right) + \left(1 - \alpha\right) \left(4 - 2 \alpha\right)$ | $\alpha \left(\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right)$ |
|
| (1, 7) | $1 - \alpha$ | (A, 3H, A, 2H, A, 2A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{8 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + 2\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [8, 9, 10] and will selfish mine on [8, 9, 10] | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [8, 9, 10] and will selfish mine on [8, 9, 10] unless recovers [1, 5] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{1 + \frac{\frac{6 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
| committed [5, 8, 9, 10] and will selfish mine on [9, 10] | $2 \alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 4\right)$ |
|
| committed [5, 8, 9, 10] and will selfish mine on [9, 10] unless recovers [1] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(2 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [5, 8, 9, 10] and will selfish mine on [9, 10] unless recovers [1] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(5 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(2 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, 4A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [8, 9, 10, 11] and will selfish mine on [8, 9, 10, 11] | $\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right)$ |
|
| committed [5, 8, 9, 10, 11] and will selfish mine on [9, 10, 11] | $2 \alpha + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 5\right)$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [5, 8, 9, 10, 11] and will selfish mine on [9, 10, 11] | $2 \alpha + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 5\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\left(1 - \alpha\right) \left(\frac{3 \alpha}{1 - 2 \alpha} + 4\right) + 2$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, 3A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [8, 9, 10] and will selfish mine on [9, 10] | $\alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [8, 9, 10] and will selfish mine on [9, 10] unless recovers [1, 5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
| committed [5, 8, 9, 10] and will selfish mine on [10] | $4 - \alpha$ |
|
| committed [5, 8, 9, 10] and will selfish mine on [10] unless recovers [1] | $4 - \alpha$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [5, 8, 9, 10] and will selfish mine on [10] unless recovers [1] | $4 - \alpha$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, 3A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] unless recovers [1, 5] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{8 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] unless recovers [1, 5] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{8 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, 2A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 2$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, 2A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, 2A, H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(4 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)$ |
|
| (2, 7) | $2 - \alpha$ | genesis | $0$ | $0$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha} + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (2, 7) | $2 - \alpha$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, 2A, H) | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, 2A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [8, 9, 11] and will selfish mine on [9, 11] | $\alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [8, 9, 11] and will selfish mine on [9, 11] unless recovers [1, 5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
| committed [5, 8, 9, 11] and will selfish mine on [11] | $4 - \alpha$ |
|
| committed [5, 8, 9, 11] and will selfish mine on [11] unless recovers [1] | $4 - \alpha$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [5, 8, 9, 11] and will selfish mine on [11] unless recovers [1] | $4 - \alpha$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, 2A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{2 \alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{2 \alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, A, 2A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{8 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + 2\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [9, 10] and will selfish mine on [9, 10] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [9, 10] and will selfish mine on [9, 10] unless recovers [5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
| committed [9, 10] and will selfish mine on [9, 10] unless recovers [1, 5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [9, 10] and will selfish mine on [9, 10] unless recovers [5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, A, A, H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 2$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, A, H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right)$ | $\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, A, H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, A, H) | $\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, A, H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(4 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)$ |
|
| (1, 9) | $1 - \alpha$ | (A, 3H, A, 2H, A, H, A) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)$ |
|
| (2, 7) | $2 - \alpha$ | genesis | $0$ | $0$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha} + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (2, 7) | $2 - \alpha$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, A, H, A) | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{2 \alpha}{1 - \alpha}\right)$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [8, 10, 11] and will selfish mine on [10, 11] | $\alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [8, 10, 11] and will selfish mine on [10, 11] unless recovers [1, 5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(\alpha + \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
| committed [5, 8, 10, 11] and will selfish mine on [11] | $4 - \alpha$ |
|
| committed [5, 8, 10, 11] and will selfish mine on [11] unless recovers [1] | $4 - \alpha$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [5, 8, 10, 11] and will selfish mine on [11] unless recovers [1] | $4 - \alpha$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 2$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{2 \alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{2 \alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, H, A, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, A, 2H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, 2H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, 3H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, A, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}\right)$ | $\alpha \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, A, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, A, H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right)\right)$ | $\alpha \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, A, H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, A, H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [1, 5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 2H, A, 2H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}}$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, 2H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 3H, A, 2H, A, 3H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}}$ |
|
Best bounds plotted together:
(A, 3H, A, 3H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 3H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right)\right) - \alpha + 1\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best bounds plotted together:
(A, 3H, A, 3H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 3H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}\right)$ | $\alpha \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
| (1, 8) | $1 - \alpha$ | (A, 3H, A, 3H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right)\right)$ | $\alpha \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (1, 8) | $1 - \alpha$ | (A, 3H, A, 3H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right)\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 3H, A, 3H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 3H, 2A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{8 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - 2 \alpha + 2\right)$ |
|
| (1, 8) | $1 - \alpha$ | (A, 3H, A, 3H, A, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [9, 10] and will selfish mine on [9, 10] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [9, 10] and will selfish mine on [9, 10] unless recovers [5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
| committed [9, 10] and will selfish mine on [9, 10] unless recovers [1, 5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [9, 10] and will selfish mine on [9, 10] unless recovers [5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best bounds plotted together:
(A, 3H, A, 3H, 3A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] unless recovers [1, 5] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{8 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] unless recovers [1, 5] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{8 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1$ |
|
Best bounds plotted together:
(A, 3H, A, 3H, 2A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 2$ |
|
Best bounds plotted together:
(A, 3H, A, 3H, A, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [1, 5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best bounds plotted together:
(A, 3H, A, 3H, A, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 3H, A, 3H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 3H, A, H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 3H, A, 3H, A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 2$ |
|
Best bounds plotted together:
(A, 3H, A, 3H, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best bounds plotted together:
(A, 3H, A, 3H, A, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 3H, A, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 3H, A, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 3H, A, 3H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 3H, A, H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}}$ |
|
Best bounds plotted together:
(A, 3H, A, 3H, A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 3H, A, 3H, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}}$ |
|
Best bounds plotted together:
(A, 3H, A, 4H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 4H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) - \alpha + 1\right)$ | $\alpha \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best bounds plotted together:
(A, 3H, A, 4H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 4H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
| (1, 9) | $1 - \alpha$ | (A, 3H, A, 4H, A) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (1, 9) | $1 - \alpha$ | (A, 3H, A, 4H, A) | $- \alpha \left(1 - \alpha\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 3H, A, 4H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [1, 5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [5] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(4 \alpha + \left(1 - \alpha\right) \left(4 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best bounds plotted together:
(A, 3H, A, 4H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 3H, A, 4H, A, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 3H, A, 4H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}}$ |
|
Best bounds plotted together:
(A, 3H, A, 5H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 3H, A, 5H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}}$ |
|
Best bounds plotted together:
(A, 3H, A, 5H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 3H, A, 6H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}}$ |
|
Best bounds plotted together:
(A, 4H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(\frac{4 \alpha^{3}}{\alpha^{2} - \alpha + 1} + \frac{\alpha^{2} \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{1}{\alpha^{2} - \alpha + 1}\right)}{\alpha^{2} - \alpha + 1} + \frac{\left(1 - \alpha\right)^{2} \left(\frac{- \alpha^{2} + \alpha + 1}{\alpha^{2} - \alpha + 1} + 1.5\right)}{\alpha^{2} - \alpha + 1}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right)$ | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\alpha \left(\frac{4 \alpha^{3}}{\alpha^{2} - \alpha + 1} + \frac{\alpha^{2} \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{1}{\alpha^{2} - \alpha + 1}\right)}{\alpha^{2} - \alpha + 1} + \frac{\left(1 - \alpha\right)^{2} \left(\frac{- \alpha^{2} + \alpha + 1}{\alpha^{2} - \alpha + 1} + 1.5\right)}{\alpha^{2} - \alpha + 1}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1\right)$ |
|
Best bounds plotted together:
(A, 4H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A) | $\alpha \left(\frac{4 \alpha^{3}}{\alpha^{2} - \alpha + 1} + \frac{\alpha^{2} \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{1}{\alpha^{2} - \alpha + 1}\right)}{\alpha^{2} - \alpha + 1} + \frac{\left(1 - \alpha\right)^{2} \left(\frac{- \alpha^{2} + \alpha + 1}{\alpha^{2} - \alpha + 1} + 1.5\right)}{\alpha^{2} - \alpha + 1}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)$ | $\alpha \left(\frac{4 \alpha^{3}}{\alpha^{2} - \alpha + 1} + \frac{\alpha^{2} \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{1}{\alpha^{2} - \alpha + 1}\right)}{\alpha^{2} - \alpha + 1} + \frac{\left(1 - \alpha\right)^{2} \left(\frac{- \alpha^{2} + \alpha + 1}{\alpha^{2} - \alpha + 1} + 1.5\right)}{\alpha^{2} - \alpha + 1}\right) + \left(1 - \alpha\right) \left(\alpha \left(2 - \alpha\right) - \alpha + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)\right)\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A) | $\alpha \left(\frac{4 \alpha^{3}}{\alpha^{2} - \alpha + 1} + \frac{\alpha^{2} \left(1 - \alpha\right) \left(\frac{5}{2} + \frac{1}{\alpha^{2} - \alpha + 1}\right)}{\alpha^{2} - \alpha + 1} + \frac{\left(1 - \alpha\right)^{2} \left(\frac{- \alpha^{2} + \alpha + 1}{\alpha^{2} - \alpha + 1} + 1.5\right)}{\alpha^{2} - \alpha + 1}\right) + \left(1 - \alpha\right) \left(- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right) - \alpha\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 4H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A, H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right)$ | $\alpha \left(2 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A, H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(2 - \alpha\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A, H) | $\alpha \left(2 - \alpha\right) + \left(1 - \alpha\right) \left(\alpha \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha}\right)\right) - \alpha + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)\right)$ |
|
Best bounds plotted together:
(A, 4H, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A, 2H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) + \alpha \left(3 - \alpha\right)\right)$ | $\alpha \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha}\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A, 2H) | $\alpha \left(\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha}\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha\right)$ |
|
Best bounds plotted together:
(A, 4H, A, 2H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A, 2H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(3 - \alpha\right)$ | $\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
| (1, 8) | $1 - \alpha$ | (A, 4H, A, 2H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}\right)\right)$ | $\alpha \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + \frac{\alpha}{1 - \alpha} + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A, 2H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(3 - \alpha\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A, 2H, A) | $\alpha \left(\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
Best bounds plotted together:
(A, 4H, A, 2H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A, 2H, 2A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(2 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}\right)$ | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 2\right)$ |
|
| (1, 8) | $1 - \alpha$ | (A, 4H, A, 2H, A, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
| (3, 5) | $3 - \alpha$ | genesis | $0$ | $0$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [9, 10] and will selfish mine on [9, 10] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [9, 10] and will selfish mine on [9, 10] unless recovers [1, 6] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(\frac{11}{2} + \frac{3 + \frac{\frac{8 \left(1 - \alpha\right)}{\alpha} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{3 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right) + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (3, 5) | $3 - \alpha$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A, 2H, 2A) | $\alpha \left(\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - 2 \alpha + \frac{\alpha}{1 - \alpha} + 2\right)$ |
|
Best bounds plotted together:
(A, 4H, A, 2H, 3A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] | $\left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right)$ |
|
| committed [9, 10, 11] and will selfish mine on [9, 10, 11] unless recovers [1, 6] | $\frac{\left(-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{8 \left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{\left(1 - \alpha\right) \left(2 + \frac{\frac{- \frac{8 \left(1 - \alpha\right)^{4}}{\alpha^{4}} + \frac{8 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}} + \frac{2 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{2 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{2 \left(2 \alpha - 1\right) \left(\frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} - \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}\right) \left(- \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{4}}{\alpha^{4}}}$ |
|
| committed [6, 9, 10, 11] and will selfish mine on [10, 11] | $2 \alpha + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 4\right)$ |
|
| committed [6, 9, 10, 11] and will selfish mine on [10, 11] unless recovers [1] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(2 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [6, 9, 10, 11] and will selfish mine on [10, 11] unless recovers [1] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(6 \alpha + \left(1 - \alpha\right) \left(6 + \frac{2 + \frac{\frac{6 \left(1 - \alpha\right)}{\alpha} - \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{2 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{\left(2 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{\frac{- \frac{6 \left(1 - \alpha\right)^{3}}{\alpha^{3}} + \frac{6 \left(1 - \alpha\right)^{4}}{\alpha^{4}}}{1 - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}\right)\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{3}}{\alpha^{3}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \left(1 - \alpha\right) \left(\frac{2 \alpha}{1 - 2 \alpha} + 3\right) + 1$ |
|
Best bounds plotted together:
(A, 4H, A, 2H, 2A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + \frac{\alpha}{1 - \alpha} + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + \frac{\alpha}{1 - \alpha} + 2$ |
|
Best bounds plotted together:
(A, 4H, A, 2H, A, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [6] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [1, 6] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [6] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best bounds plotted together:
(A, 4H, A, 2H, A, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 4H, A, 2H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A, 2H, A, H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + \frac{\alpha}{1 - \alpha} + 2\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} + \frac{\alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 4H, A, 2H, A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + \frac{\alpha}{1 - \alpha} + 2$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha + \frac{\alpha}{1 - \alpha} + 2$ |
|
Best bounds plotted together:
(A, 4H, A, 2H, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}}$ |
|
Best bounds plotted together:
(A, 4H, A, 2H, A, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A, 2H, A, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A, 2H, A, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 4H, A, 2H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A, 2H, A, H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best bounds plotted together:
(A, 4H, A, 2H, A, H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 4H, A, 2H, A, 2H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}}$ |
|
Best bounds plotted together:
(A, 4H, A, 3H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A, 3H) | $- \alpha \left(1 - \alpha\right) + \alpha \left(- \alpha \left(1 - \alpha\right) - \alpha + 1\right)$ | $\alpha \left(\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}}$ |
|
Best bounds plotted together:
(A, 4H, A, 3H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A, 3H, A) | $- \alpha \left(1 - \alpha\right) + \alpha \left(\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}\right)$ | $\alpha \left(\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + \frac{\alpha}{1 - \alpha}\right)$ |
|
| (1, 9) | $1 - \alpha$ | (A, 4H, A, 3H, A) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| (1, 9) | $1 - \alpha$ | (A, 4H, A, 3H, A) | $- \alpha \left(1 - \alpha\right)$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} + \frac{\alpha^{2}}{\left(1 - \alpha\right)^{2}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 4H, A, 3H, 2A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] | $\left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [6] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [1, 6] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(7 \alpha + \left(1 - \alpha\right) \left(6 + \frac{4 + \frac{\frac{10 \left(1 - \alpha\right)}{\alpha} - \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}}}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{4 \left(1 - \alpha\right)}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{10 \left(1 - \alpha\right)^{5}}{\alpha^{5}} + \frac{10 \left(1 - \alpha\right)^{6}}{\alpha^{6}}}{1 - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{5}}{\alpha^{5}}}$ |
|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| committed [10, 11] and will selfish mine on [10, 11] unless recovers [6] | $\frac{\left(-1 + \frac{1 - \alpha}{\alpha}\right) \left(3 \alpha + \left(1 - \alpha\right) \left(\frac{7}{2} + \frac{1 + \frac{\frac{4 \left(1 - \alpha\right)}{\alpha} - \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}}}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha}}{2 \left(1 - \frac{1 - \alpha}{\alpha}\right) \left(2 \alpha - 1\right)}\right)\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{\left(1 - \alpha\right) \left(\frac{3}{2} + \frac{\frac{- \frac{4 \left(1 - \alpha\right)^{2}}{\alpha^{2}} + \frac{4 \left(1 - \alpha\right)^{3}}{\alpha^{3}}}{1 - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}} + \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}{2 \left(2 \alpha - 1\right) \left(\frac{1 - \alpha}{\alpha} - \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}\right) \left(- \frac{1 - \alpha}{\alpha} + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}\right)}{-1 + \frac{\left(1 - \alpha\right)^{2}}{\alpha^{2}}}$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}} + \frac{\alpha}{1 - \alpha} + \left(1 - \alpha\right) \left(\frac{\alpha}{1 - 2 \alpha} + 2\right)$ |
|
Best bounds plotted together:
(A, 4H, A, 3H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha}{1 - \alpha}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} + \frac{\alpha}{1 - \alpha}$ |
|
Best bounds plotted together:
(A, 4H, A, 3H, A, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 4H, A, 3H, A, H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}}$ |
|
Best bounds plotted together:
(A, 4H, A, 4H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Wait | $0$ | (A, 4H, A, 4H) | $- \alpha \left(1 - \alpha\right)$ | $\alpha \left(\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1\right) + \left(1 - \alpha\right) \left(\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}} - \alpha\right)$ |
|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{7}}{\left(1 - \alpha\right)^{7}} + \frac{\alpha^{4}}{\left(1 - \alpha\right)^{4}}$ |
|
Best bounds plotted together:
(A, 4H, A, 4H, A)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{6}}{\left(1 - \alpha\right)^{6}} + \frac{\alpha^{3}}{\left(1 - \alpha\right)^{3}} - \alpha + 1$ |
|
Best bounds plotted together:
(A, 4H, A, 5H)
Action bounds:
| Action | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|
Commitment bounds:
| Commitment | Lower Bound | Plot |
|---|
Lemma bounds:
| Lemma | Lower Bound | Upper Bound | Plot |
|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}}$ |
|
Best Lower Bound:
| Action, Commitment or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Capitulate | $0$ | genesis | $0$ |
|
Best Upper Bound:
| Action or Lemma | Immediate Reward | Subsequent State | Lower Bound | Upper Bound | Plot |
|---|---|---|---|---|---|
| Lemma G.8 | $\frac{\alpha^{8}}{\left(1 - \alpha\right)^{8}} + \frac{\alpha^{5}}{\left(1 - \alpha\right)^{5}}$ |
|
Best bounds plotted together:
