Heads-up Limit Hold’em Poker is Solved

Poker is a family of games that exhibit imperfect information, where players do not have full knowledge of past events. Whereas many perfect informa-tion 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 es-sentially 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 solv-ing extensive-form games orders of magnitude larger than previously possible. Games have been intertwined with the earliest developments in computation, game theory, and artificial intelligence (AI). At the very conception of computing, Babbage had detailed plans for an “automaton ” capable of playing tic-tac-toe and dreamt of his Analytical Engine playing chess (1). Both Turing (2) and Shannon (3) — on paper and in hardware, respectively — de-veloped programs to play chess as a means of validating early ideas in computation and AI. For more than a half century, games have continued to act as testbeds for new ideas and the result