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.

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Game description[edit]

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

• Each player antes 1.
• Each player is dealt one of the three cards, and the third is put aside unseen.
• Player one can check or bet 1.
  • If player one checks then player two can check or bet 1.
    • If player two checks there is a showdown for the pot of 2 (i.e. the higher card wins 1 from the other player).
    • If player two bets then player one can fold or call.
      • If player one folds then player two takes the pot of 3 (i.e. winning 1 from player 1).
      • If player one calls there is a showdown for the pot of 4 (i.e. the higher card wins 2 from the other player).
  • If player one bets then player two can fold or call.
    • If player two folds then player one takes the pot of 3 (i.e. winning 1 from player 2).
    • If player two calls there is a showdown for the pot of 4 (i.e. the higher card wins 2 from the other player).

Optimal strategy[edit]

The game has a mixed-strategyNash 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.

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• The charts below show the Nash Equilibrium strategies for two player push-or-fold NLHE. This is a simplified game where the SB is only allowed to go all-in or fold, and the BB can either call or fold when facing a shove.

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 (otherwise he checks; if the other player bets, he should always fold). When having a King, he should bet with the probability of ${\displaystyle 3\alpha }$ (otherwise he checks; if the other player bets, he should always call). 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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The second player has a single equilibrium strategy: Always betting or calling when having a King; when having a Queen, checking if possible, otherwise calling with the probability of 1/3; when having a Jack, never calling and betting with the probability of 1/3.

Generalized versions[edit]

In addition to the basic version invented by Kuhn, other versions appeared adding bigger deck, more players, betting rounds, etc., increasing the complexity of the game.

3-player Kuhn Poker[edit]

A variant for three players was introduced in 2010 by Nick Abou Risk and Duane Szafron. In this version, the deck includes four cards (adding a ten card), from which three are dealt to the players; otherwise, the basic structure is the same: while there is no outstanding bet, a player can check or bet, with an outstanding bet, a player can call or fold. If all players checked or at least one player called, the game proceeds to showdown, otherwise, the betting player wins.

A family of Nash equilibria for 3-player Kuhn poker is known analytically, which makes it the largest game with more than two players with analytic solution.^[1] The family is parameterized using 4–6 parameters (depending on the chosen equilibrium). In all equilibria, player 1 has a fixed strategy, and he always checks as the first action; player 2's utility is constant, equal to –1/48 per hand. The discovered equilibrium profiles show an interesting feature: by adjusting a strategy parameter ${\displaystyle \beta }$ (between 0 and 1), player 2 can freely shift utility between the other two players while still remaining in equilibrium; player 1's utility is equal to ${\displaystyle -\frac{1+2\beta }{48}}$ (which is always worse than player 2's utility), player 3's utility is ${\displaystyle \frac{1+\beta }{24}}$.

It is not known if this equilibrium family covers all Nash equilibria for the game.

References[edit]

• Kuhn, H. W. (1950). 'Simplified Two-Person Poker'. In Kuhn, H. W.; Tucker, A. W. (eds.). Contributions to the Theory of Games. 1. Princeton University Press. pp. 97–103.
• James Peck. 'Perfect Bayesian Equilibrium'(PDF). Ohio State University. Retrieved 2 September 2016.^:19–29

1. ^Szafron, Duane; Gibson, Richard; Sturtevant, Nathan (May 2013). 'A Parameterized Family of Equilibrium Profiles forThree-Player Kuhn Poker'(PDF). In Ito; Jonker; Gini; Shehory (eds.). Proceedings of the 12th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2013). Saint Paul, Minnesota, USA.

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Calculation of equilibrium ranges in mixed unrestricted strategies include 3-way pots and all ties.

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