Heads-up limit Hold'em Poker is solved

Abstract

Poker is a family of games that exhibit imperfect information, where players do not have full knowledge of past events. While 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. In this paper, we announce that the smallest variant of poker in-play, 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 significant advantage. This result was enabled by a new algorithm, CFR+, which is capable of solving extensive-form games three orders of magnitude larger than previously possible. This paper is an extended version of the original 2015 Science article,9 with additional results showing Cepheus' in-game performance against computer and human opponents.