Kuhn poker

Kuhn poker is an extremely simplified form of poker developed by Harold W. Kuhn as a simple model zero-sum two-player imperfect-information game, amenable to a complete game-theoretic analysis. In Kuhn poker, the deck includes only three playing cards, for example a King, Queen, and Jack. One card is dealt to each player, which may place bets similarly to a standard poker. If both players bet or both players pass, the player with the higher card wins, otherwise, the betting player wins.

In conventional poker terms, a game of Kuhn poker proceeds as follows:

The game has a mixed-strategy Nash equilibrium; when both players play equilibrium strategies, the first player should expect to lose at a rate of −1/18 per hand (as the game is zero-sum, the second player should expect to win at a rate of +1/18). There is no pure-strategy equilibrium.

Kuhn demonstrated there are infinitely many equilibrium strategies for the first player, forming a continuum governed by a single parameter. In one possible formulation, player one freely chooses the probability ${\displaystyle \alpha \in [0,1/3]}$ with which he will bet when having a Jack. Then, when having a King, he should bet with the probability of ${\displaystyle 3\alpha }$; he should always check when having a Queen, and if the other player bets after this check, he should call with the probability of ${\displaystyle \alpha +1/3}$.

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