---
title: "Solving Heads-Up Limit Texas Hold'em"
authors:
  - "Oskari Tammelin"
  - "Neil Burch"
  - "Michael Johanson"
  - "Michael H. Bowling"
doi: ""
url: "https://ijcai.org/proceedings/2015/97"
pdf: "https://ijcai.org/proceedings/2015/0097.pdf"
venue: "Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence"
year: 2015
pages: "645-651"
track: "Main Track"
keywords:
---

# Solving Heads-Up Limit Texas Hold'em

Oskari Tammelin, Neil Burch, Michael Johanson, Michael H. Bowling

*Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence*, pages 645-651.

## Abstract

Cepheus is the first computer program to essentially solve a game of imperfect information that is played competitively by humans. The game it plays is heads-up limit Texas hold'em poker, a game with over 10^14 information sets, and a challenge problem for artificial intelligence for over 10 years. Cepheus was trained using a new variant of Counterfactual Regret Minimization (CFR), called CFR+, using 4800 CPUs running for 68 days. In this paper we describe in detail the engineering details required to make this computation a reality. We also prove the theoretical soundness of CFR+ and its component algorithm, regret-matching+. We further give a hint towards understanding the success of CFR+ by proving a tracking regret bound for this new regret matching algorithm. We present results showing the role of the algorithmic components and the engineering choices to the success of CFR+.