Kuhn poker
   HOME

TheInfoList



OR:

Kuhn poker is an extremely simplified form of
poker Poker is a family of comparing card games in which players wager over which hand is best according to that specific game's rules. It is played worldwide, however in some places the rules may vary. While the earliest known form of the game w ...
developed by
Harold W. Kuhn Harold William Kuhn (July 29, 1925 – July 2, 2014) 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 ...
as a simple model
zero-sum Zero-sum game is a mathematical representation in game theory and economic theory of a situation which involves two sides, where the result is an advantage for one side and an equivalent loss for the other. In other words, player one's gain is e ...
two-player imperfect-information game, amenable to a complete
game-theoretic Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applic ...
analysis. In Kuhn poker, the deck includes only three
playing cards A playing card is a piece of specially prepared card stock, heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic that is marked with distinguishing motifs. Often the front (face) and back of each card has a fi ...
, 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 Check or cheque, may refer to: Places * Check, Virginia Arts, entertainment, and media * ''Check'' (film), a 2021 Indian Telugu-language film * ''The Checks'' (episode), a 1996 TV episode of ''Seinfeld'' Games and sports * Check (chess), a thr ...
or
bet Black Entertainment Television (acronym BET) is an American basic cable channel targeting African-American audiences. It is owned by the CBS Entertainment Group unit of Paramount Global via BET Networks and has offices in New York City, Los A ...
1. **If player one checks then player two can check or bet 1. ***If player two checks there is a
showdown A showdown is a duel. The term may also refer to: Places * Showdown Ski Area, in Montana, United States Books * ''Showdown'' (Amado novel), a 1984 novel by Jorge Amado * ''Showdown'' (Dekker novel), a 2006 novel by Ted Dekker * ''Showdown'' (F ...
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 Call or Calls may refer to: Arts, entertainment, and media Games * Call, a type of betting in poker * Call, in the game of contract bridge, a bid, pass, double, or redouble in the bidding stage Music and dance * Call (band), from Lahore, Paki ...
. ****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 In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...
; 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 Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
\alpha \isin , 1/3/math> 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 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 \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. 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 \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 -\frac (which is always worse than player 2's utility), player 3's utility is \frac. It is not known if this equilibrium family covers all Nash equilibria for the game.


References

* *


External links


Effective Short-Term Opponent Exploitation in Simplified Poker
{{game theory Non-cooperative games Poker variants 20th-century card games Games and sports introduced in 1950