Heads-up Limit Hold'em Poker is Solved

Michael Bowling, Neil Burch, Michael Johanson, and Oskari Tammelin. Heads-up limit hold'em poker is solved. In 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 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.


In this paper, we announced that we have essentially weakly solved the game of heads-up limit Texas hold'em. This marks the first time that any game of imperfect information that is played by humans has been solved. To accomplish this result, we used a new variant of CFR called CFR+, invented by Oskari Tammelin, which converges much more quickly than the standard algorithm.

"Cepheus" is the name of our computer program that uses this solution. In addition to releasing the source code for the solver, we also have Cepheus online on this page, where you can play against it, or inspect its strategy to find out what it would do at any decision point in the game.

This paper focusses on the effort required to reach the milestone of solving the game. For more information on CFR+, including proofs of convergence and empirical results on smaller games comparing CFR and CFR+, read the companion paper that we published at IJCAI 2015: Solving Heads-up Limit Texas Hold'em.


Heads-up limit Texas hold'em was the first imperfect information game played competitively by humans to be solved, and our paper about reaching this milestone received worldwide media attention. Searching for "cepheus poker" will turn up hits. Here are some of the notable ones:


  • Our followup CFR+ paper, which goes into much more detail about the CFR+ algorithm, including theoretical convergence proofs, and engineering details that made solving HULHE possible.
  • Cepheus Website, where you can play against Cepheus and query its strategy at any decision point.
  • Oskari Tammelin's CFR page. Oskari developed the CFR+ algorithm, and was part of our team for solving HULHE. His website has a fantastic demo of CFR algorithms, where you can edit the code and watch the strategies evolve in real time.


  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"