Jail BlackJack (under construction)
Summary of the game
The name of the game is due to its alleged popularity at the correctional institutions of the Soviet Union. Decks of playing cards were forbidden there because the cards were synonymous with gambling. This game does not require the cards, the fingers being used instead.
The game is for two players. The goal is the same as in blackjack - to get the highest possible number of points not exceeding 21. A player with more than 21 points loses immediately (is busted). If, at the end of the game, neither player is busted, the one with the higher score wins. If the two scores are the same, it's a tie.
The players acquire their points in turn. One player goes first. Once the first player's final score has been determined, the process of determining the second player's score begins. If the first player is busted, there is no need for the second player to acquire any points; the second player is the winner in that case. A player acquires points by starting with zero and going through one or several rounds of play. A round of play involves the two players showing simultaneously any number of fingers, from zero to ten. The combined number of shown fingers gets added to the running score of the player whose turn it is. If the player is busted, the game is over. Otherwise, the player has two options: to stop, in which case the current score becomes the player's final score, or to keep going, in which case another round is played out. Naturally, if a player has 12 or more points, the player must stop. Otherwise, in the next round the opponent would show 10 fingers and bust the player.
One additional rule: if the first player hits blackjack, i.e. scores exactly 21, the second player is not given a chance to try accomplish the same feat and tie the game. The game gets declared to be won by the first player. Without that rule, the second player would have enjoyed a slight advantage. With the rule in place and both players exercising optimal strategy, the chances are almost even: 50.001% for the first player and 49.999% for the second player.
It is important to note that a skillful player repeatedly facing the same human opponent often achieves a much better result than a computer that follows the optimal strategy. A skillful player is able to perceive certain patterns in the opponent's behavior that are tantamount to deviation from the optimal strategy by the opponent. The skillful player then takes a calculated risk and deviates from the optimal strategy as well, in order to punish the opponent for the perceived deviations.
We assume that in each game the winner gets the bank. In a tie, the two players split the bank evenly. In real competitions, the contestants play longer matches. One typical match arrangement is to play until one of the players wins 10 games. The ties are not counted. In each game, the loser of the previous game goes first. If the previous game was a tie, the second player in that game goes first.
Values of various positions
1st player points | 2nd player expectation | Absent the "21 wins" rule |
---|---|---|
0 | 50.00% | > 50% |
1 | 50.08% | |
2 | 50.72% | |
3 | 50.51% | |
4 | 50.01% | |
5 | 49.13% | |
6 | 48.40% | |
7 | 47.66% | |
8 | 47.43% | |
9 | 47.50% | |
10 | 48.54% | |
11 | 51.06% | |
12 | 79.16% | |
13 | 64.50% | |
14 | 55.67% | |
15 | 47.73% | |
16 | 41.89% | |
17 | 28.09% | |
18 | 21.31% | |
19 | 13.32% | |
20 | 6.82% | |
21 | 0.00% | 1.83% |
22 or more | 100% |
One interesting observation: as can be seen from the above table, the scores between 15 and 21 for the 1st player favor that player. For example, if the 1st player scores 15, the 2nd player's expectations are 47.73%, below 1/2.
2nd player's turn | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |
---|---|---|---|---|---|---|---|---|---|---|
0 | 79.16% | 64.50% | 55.67% | 47.73% | 41.89% | 28.09% | 21.31% | 13.32% | 6.82% | 1.83% |
1 | 79.05% | 64.41% | 55.58% | 47.60% | 41.72% | 27.95% | 21.28% | 13.32% | 6.83% | 1.84% |
2 | 79.05% | 63.69% | 54.13% | 45.59% | 39.29% | 26.40% | 20.14% | 12.70% | 6.61% | 1.80% |
3 | 81.74% | 64.87% | 54.21% | 45.70% | 39.22% | 26.38% | 19.90% | 12.55% | 6.50% | 1.77% |
4 | 81.64% | 70.45% | 57.33% | 45.91% | 39.50% | 26.59% | 19.89% | 12.49% | 6.45% | 1.76% |
5 | 80.96% | 70.45% | 62.87% | 50.88% | 40.25% | 26.84% | 20.04% | 12.52% | 6.45% | 1.76% |
6 | 80.14% | 69.17% | 62.87% | 55.59% | 50% | 27.18% | 20.34% | 12.85% | 6.55% | 1.78% |
7 | 79.10% | 67.47% | 62.22% | 55.59% | 50.13% | 30.81% | 21.23% | 13.37% | 6.77% | 1.83% |
8 | 77.73% | 65.18% | 60% | 55.56% | 50% | 37.72% | 27.61% | 13.88% | 7.19% | 1.92% |
9 | 75.79% | 62.22% | 4/7 | 50% | 50% | 37.72% | 31.39% | 19.78% | 7.91% | 2.10% |
10 | 8/11 | 4/7 | 50% | 50% | 50% | 37.63% | 31.39% | 22.24% | 12.57% | 2.51% |
11 | 2/3 | 50% | 50% | 50% | 50% | 4/11 | 31.39% | 22.24% | 13.48% | 4.58% |
Note: in the above table, the numbers in the rightmost column show the 2nd player's chances in the absence of the rule declaring 21 by the 1st player to be an instantaneous win. Under the rule, in the competitive games all these position values are 0.00%.
Below, the optimal strategy is demonstrated for the 14/8 position, i.e. the 1st player having earned 14 points and the 2nd currently having 8 points. In that position, the 2nd player expects to get 60% of the bank. If the 2nd player shows "0" 40% of the time, "3" 20% of the time, and "7" 40% of the time, there is nothing the 1st player can do to reduce the expected win of the 2nd player to under 60%. At the same time, if the 1st player shows "2" 40% of the time, "4" 20% of the time, and "10" 40% of the time, there is nothing the 2nd player can do in order to win over 60% of the bank, on average.
----- | 14 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 1st player possible moves |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
8 | ----- | ----- | ----- | 40% | ----- | 20% | ----- | ----- | ----- | ----- | ----- | 40% | 1st player optimal strategy |
0 | 40% | 0.6 | 4/7 | 0.5 | 0.5 | 0 | 0 | 0.5 | 1 | 1 | 1 | 1 | |
1 | ----- | 4/7 | 0.5 | 0.5 | 0 | 0 | 0.5 | 1 | 1 | 1 | 1 | 1 | |
2 | ----- | 0.5 | 0.5 | 0 | 0 | 0.5 | 1 | 1 | 1 | 1 | 1 | 1 | |
3 | 20% | 0.5 | 0 | 0 | 0.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
4 | ----- | 0 | 0 | 0.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | |
5 | ----- | 0 | 0.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | |
6 | ----- | 0.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | |
7 | 40% | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | |
8 | ----- | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | |
9 | ----- | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | |
10 | ----- | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |