QL2, a simple reinforcement learning scheme for two-player zero-sum Markov games

Benoît Frénay, Marco Saerens

Research output: Contribution to journalArticlepeer-review


Markov games is a framework which can be used to formalise n-agent reinforcement learning (RL). Littman (Markov games as a framework for multi-agent reinforcement learning, in: Proceedings of the 11th International Conference on Machine Learning (ICML-94), 1994.) uses this framework to model two-agent zero-sum problems and, within this context, proposes the minimax-Q algorithm. This paper reviews RL algorithms for two-player zero-sum Markov games and introduces a new, simple, fast, algorithm, called QL2. QL2 is compared to several standard algorithms (Q-learning, Minimax and minimax-Q) implemented with the Q ash library written in Python. The experiments show that QL2 converges empirically to optimal mixed policies, as minimax-Q, but uses a surprisingly simple and cheap updating rule. © 2009 Elsevier B.V. All rights reserved.

Original languageEnglish
Pages (from-to)1494-1507
Number of pages14
Issue number7-9
Publication statusPublished - 2008
Externally publishedYes


  • Markov games
  • Multi-agent
  • Q-Learning
  • Reinforcement learning
  • Two-player zero-sum games


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