Kuhn poker is a simplified form ofpoker developed byHarold W. Kuhn as a simple modelzero-sum two-playerimperfect-information game, amenable to a completegame-theoretic analysis. In Kuhn poker, the deck includes only threeplaying cards, for example, a King, Queen, and Jack. Both players are dealt a card, and they may choose tobet orcheck. If both players bet or both players check, the player with the higher card wins. Otherwise, the betting player wins.

Inconventional poker terms, a game of Kuhn poker proceeds as follows:

• Each playerantes 1.
• Each player is dealt one of the three cards, and the third is put aside unseen.
• Player one cancheck orbet 1.
  • If player one checks then player two can check or bet 1.
    • If player two checks there is ashowdown for the pot of 2 (i.e. the higher card wins 1 from the other player).
    • If player two bets then player one canfold orcall.
      • 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).

The game has amixed-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 nopure-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 theprobability${\displaystyle \alpha \in [0,1/3]}$ with which they will bet when having a Jack (otherwise they check; if the other player bets, they should always fold). When having a King, they should bet with the probability of${\displaystyle 3\alpha }$ (otherwise they check; if the other player bets, they should always call). They should always check when having a Queen, and if the other player bets after this check, they should call with the probability of${\displaystyle \alpha +1/3}$.

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.

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.

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 they always check 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.

• Kuhn, H. W. (1950)."Simplified Two-Person Poker"(PDF). In Kuhn, H. W.; Tucker, A. W. (eds.).Contributions to the Theory of Games. Vol. 1. Princeton University Press. pp. 97–103.
• James Peck."Perfect Bayesian Equilibrium"(PDF). Ohio State University. Retrieved2 September 2016.^: 19–29

• Effective Short-Term Opponent Exploitation in Simplified Poker

• Non-cooperative games
• Poker variants
• 20th-century card games
• Games and sports introduced in 1950

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