Question 15.53
21. Consider the following poker game with two players, I and II. Each antes $1. Each player is dealt either a high card or a low card , with probability 1/2. Player I then folds or bets $1. If Player I bets, then Player II either folds, calls, or raises $1. Finally, if II raises, Player I either folds or calls.
Most choices by the players are rather obvious, at least to anyone who has played poker: If either player holds , that player always bets or raises if he or she gets the choice. The question remains of how often one should bluff—that is, continue to play (by calling or raising) while holding a low card in the hope that one’s opponent also holds a low card.
This poker game can be represented by the following matrix game, in which the payoffs are the expected winnings for Player I (depending on the random deal) and the dominated strategies have been eliminated.
| Player II (when holding L) | ||||
|---|---|---|---|---|
| Folds | Calls | Raises | ||
| Player I (when holding L) | Folds initially | −0.25 | 0 | 0.25 |
| Bets first, folds later | 0 | 0 | −0.25 | |
| Bets first, calls later | −0.25 | −0.25 | 0 | |
- Are there any strategies in this matrix game that a player should avoid?
- Solve this game.
- Which player is in the more favored position?
21.
(a) “Bet, then call"should be avoided by Player I.
(b) Player I ; Player II: ; value
(c) Player II