Michael Bowling, Neil Burch, Michael Johanson, and Oskari Tammelin. Heads-up Limit Hold'em Poker is Solved. Science, 347(6218):145–149, January 2015.
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 holdâem 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.
The HTML link is to Science's paywall'd site. The PDF download is our accepted manuscript.
@Article(15science, Title = "Heads-up Limit Hold'em Poker is Solved", Author = "Michael Bowling and Neil Burch and Michael Johanson and Oskari Tammelin", Journal = "Science", Month = "January", Year = "2015", Volume = "347", Number = "6218", Pages = "145--149" )