Optimal Strategy for the 18 / 11 position in Jail Blackjack
| ----- | 18 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11 | ----- | X | ----- | ----- | X/2 | X/2 | ----- | X/4 | X/2 | ----- | ----- | Z |
| 0 | X | X | 0 | 0 | 0 | 0 | 0 | 0 | 0.5 | 1 | 1 | 1 |
| 1 | ----- | 0 | 0 | 0 | 0 | 0 | 0 | 0.5 | 1 | 1 | 1 | 0 |
| 2 | ----- | 0 | 0 | 0 | 0 | 0 | 0.5 | 1 | 1 | 1 | 0 | 0 |
| 3 | X/2 | 0 | 0 | 0 | 0 | 0.5 | 1 | 1 | 1 | 0 | 0 | 0 |
| 4 | X/2 | 0 | 0 | 0 | 0.5 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| 5 | ----- | 0 | 0 | 0.5 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | X/4 | 0 | 0.5 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | X/2 | 0.5 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | Z | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | ----- | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | ----- | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Here, X = (7 - sqrt(33)) / 4 = 31.3859%; Z = 1 - 11 * X / 4 = 13.6887%
Explanatory notes:
First row - possible moves by the 1st player (blue).
First column - possible moves by the 2nd player (blue).
Second row - probability of the 1st player making the corresponding move, under the optimal strategy (yellow).
Second column - probability of the 2nd player making the corresponding move, under the optimal strategy (yellow).
The matrix elements - the value of the corresponding outcome for the 2nd player. Naturally, the value of the outcome corresponding to "0" moves by both players is the value of the position itself.