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 ...