# Kuhn poker

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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.

Poker is a family of card games that combines gambling, strategy, and different skills. All poker variants involve betting as an intrinsic part of play, and determine the winner of each hand according to the combinations of players' cards, at least some of which remain hidden until the end of the hand. Poker games vary in the number of cards dealt, the number of shared or "community" cards, the number of cards that remain hidden, and the betting procedures.

Harold William Kuhn was an American mathematician who studied game theory. He won the 1980 John von Neumann Theory Prize along with David Gale and Albert W. Tucker. A former Professor Emeritus of Mathematics at Princeton University, he is known for the Karush–Kuhn–Tucker conditions, for Kuhn's theorem, for developing Kuhn poker as well as the description of the Hungarian method for the assignment problem. Recently, though, a paper by Carl Gustav Jacobi, published posthumously in 1890 in Latin, has been discovered that anticipates by many decades the Hungarian algorithm.

In game theory and economic theory, a zero-sum game is a mathematical representation of a situation in which each participant's gain or loss of utility is exactly balanced by the losses or gains of the utility of the other participants. If the total gains of the participants are added up and the total losses are subtracted, they will sum to zero. Thus, cutting a cake, where taking a larger piece reduces the amount of cake available for others, is a zero-sum game if all participants value each unit of cake equally.

## Game description

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

The following is a glossary of poker terms used in the card game of poker. It supplements the glossary of card game terms. Besides the terms listed here, there are thousands of common and uncommon poker slang terms. This is not intended to be a formal dictionary; precise usage details and multiple closely related senses are omitted here in favor of concise treatment of the basics.

• 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).

In poker, the showdown is a situation when, if more than one player remains after the last betting round, remaining players expose and compare their hands to determine the winner or winners.

## 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 a rate of +1/18). There is no pure-strategy equilibrium.

In game theory, the Nash equilibrium, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.

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 ${\displaystyle \alpha +1/3}$.

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

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

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.

## Related Research Articles

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## References

• 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.