Research Article

Heads-up limit hold’em poker is solved

Science  09 Jan 2015:
Vol. 347, Issue 6218, pp. 145-149
DOI: 10.1126/science.1259433

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I'll see your program and raise you mine

One of the fundamental differences between playing chess and two-handed poker is that the chessboard and the pieces on it are visible throughout the entire game, but an opponent's cards in poker are private. This informational deficit increases the complexity and the uncertainty in calculating the best course of action—to raise, to fold, or to call. Bowling et al. now report that they have developed a computer program that can do just that for the heads-up variant of poker known as Limit Texas Hold 'em (see the Perspective by Sandholm).

Science, this issue p. 145; see also p. 122


Poker is a family of games that exhibit imperfect information, where players do not have full knowledge of past events. Whereas many perfect-information games have been solved (e.g., Connect Four and checkers), no nontrivial imperfect-information game played competitively by humans has previously been solved. Here, we announce that heads-up limit Texas hold’em is now essentially weakly solved. Furthermore, this computation formally proves the common wisdom that the dealer in the game holds a substantial advantage. This result was enabled by a new algorithm, CFR+, which is capable of solving extensive-form games orders of magnitude larger than previously possible.

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