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.

Game description

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

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 the same rate). 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 $\alpha \in [0,1/3]$ with which he will bet when having a Jack. Then, when having a King, he should bet with the probability of $3\alpha$ ; he should always check when having a Queen, and if the other player bets after this check, he should call with the probability of $\alpha +1/3$ .

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.

Complete tree of Kuhn poker including probabilities for mixed-strategy Nash equilibrium. Dotted lines mark subtrees for dominated strategies.

References

Kuhn, H. W. (1950). "Simplified Two-Person Poker". In Kuhn, H. W.; Tucker, A. W. 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

External links

Effective Short-Term Opponent Exploitation in Simplified Poker

Topics in game theory

Definitions	Normal-form game Extensive-form game Escalation of commitment Graphical game Cooperative game Succinct game Information set Hierarchy of beliefs Preference

Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian Nash equilibrium Perfect Bayesian equilibrium Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Gibbs equilibrium Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium

Strategies	Dominant strategies Pure strategy Mixed strategy Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy

Classes of games	Symmetric game Perfect information Simultaneous game Sequential game Repeated game Signaling game Screening game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global games Strictly determined game Potential game

Games	Prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Rock-paper-scissors Pirate game Dictator game Public goods game Blotto games War of attrition El Farol Bar problem Fair division Fair cake-cutting Cournot game Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Prisoners and hats puzzle Trust game Princess and monster game Monty Hall problem Rendezvous problem

Theorems	Minimax theorem Nash's theorem Purification theorem Folk theorem Revelation principle Arrow's impossibility theorem

Key figures	Albert W. Tucker Amos Tversky Ariel Rubinstein Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin Jean Tirole Jean-François Mertens John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Thomas Schelling William Vickrey

See also	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition List of game theorists List of games in game theory No-win situation Topological game Tragedy of the commons Tyranny of small decisions

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