Artificial intelligence has seen several breakthroughs in recent years, with games often serving as milestones. A common feature of these games is that players have perfect information. Poker, the quintessential game of imperfect information, is a long-standing challenge problem in artificial intelligence. We introduce DeepStack, an algorithm for imperfect-information settings. It combines recursive reasoning to handle information asymmetry, decomposition to focus computation on the relevant decision, and a form of intuition that is automatically learned from self-play using deep learning. In a study involving 44,000 hands of poker, DeepStack defeated, with statistical significance, professional poker players in heads-up no-limit Texas hold'em. The approach is theoretically sound and is shown to produce strategies that are more difficult to exploit than prior approaches.

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.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 -developed 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 resulting successes have marked important milestones in the progress of AI. Examples include the checkers-playing computer program Chinook becoming the first to win a world championship title against humans (4), Deep Blue defeating Kasparov in chess (5), and Watson defeating Jennings and Rutter on Jeopardy! (6). However, defeating top human players is not the same as "solving" a game -that is, computing a game-theoretically optimal solution that is incapable of losing against any opponent in a fair game. Notable milestones in the advancement of AI have been achieved through solving games such as Connect Four (7) and checkers (8).Every nontrivial game played competitively by humans that has been solved to date is a perfect-information game (9). In perfect-information games, all players are informed of everything that has occurred in the game before making a decision. Chess, checkers, and backgammon are examples of perfect-information games. In imperfect-information games, players do 1

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We present an algorithm which determines the outcome of an arbitrary Hex game-state by finding a winning virtual connection for the winning player. Our algorithm performs a recursive descent search of the game-tree, combining fixed and dynamic game-state virtual connection composition rules with some new Hex game-state reduction results based on move domination. The algorithm is powerful enough to solve arbitrary 7 7 game-states; in particular, we use it to determine the outcome of a 7 7 Hex game after each of the 49 possible opening moves, in each case finding an explicit proof-tree for the winning player.

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