Mathematics Of Poker

Mathematics Of Poker

Mathematical game theory was introduced by John von Neumann in the 1940s, and has since become one of the foundations of modern economics [von Neumann and Morgenstern, 1944]. Von Neumann used the game of poker as a basic model for 2-player zero-sum adversarial games, and proved the first fundamental result, the famous minimax theorem. A few years later, John Nash added results for N-player noncooperative games, for which he later won the Nobel Prize [Nash, 1950]. Many decision problems can be modeled using game theory, and it has been employed in a wide variety of domains in recent years.

Of particular interest is the existence of optimal solutions, or Nash equilibria. An optimal solution provides a randomized mixed strategy, basically a recipe of how to play in each possible situation. Using this strategy ensures that an agent will obtain at least the game-theoretic value of the game, regardless of the opponent’s strategy. Unfortunately, finding exact optimal solutions is limited to relatively small problem sizes, and is not practical for most real domains.

This paper explores the use of highly abstracted mathematical models which capture the most essential properties of the real domain, such that an exact solution to the smaller problem provides a useful approximation of an optimal strategy for the real domain. The application domain used is the game of poker, specifically Texas Hold’em, the most popular form of casino poker and the poker variant used to determine the world champion at the annual World Series of Poker.

Due to the computational limitations involved, only simplified poker variations have been solved in the past (e.g. [Kuhn, 1950; Sakaguchi and Sakai, 1992]). While these are of theoretical interest, the same methods are not feasible for real games, which are too large by many orders of magnitude ([Koller and Pfeffer, 1997]).

[Shi and Littman, 2001] investigated abstraction techniques to reduce the large search space and complexity of the problem, using a simplified variant of poker. [Takusagawa, 2000] created near-optimal strategies for the play of three specific Hold’em flops and betting sequences. [Selby, 1999] computed an optimal solution for the abbreviated game of preflop Hold’em.

Using new abstraction techniques, we have produced viable “pseudo-optimal” strategies for the game of 2-player Texas Hold’em. The resulting poker-playing programs have demonstrated a tremendous improvement in performance. Whereas the previous best poker programs were easily beaten by any competent human player, the new programs are capable of defeating very strong players, and can hold their own against world-class opposition.

Although some domain-specific knowledge is an asset in creating accurate reduced-scale models, analogous methods can be developed for many other imperfect information domains and generalized game trees. We describe a general method of problem reformulation that permits the independent solution of sub-trees by estimating the conditional probabilities needed as input for each computation.

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