Algorithms for computing strategies in two-player simultaneous move games

Branislav Bosansky*, Viliam Lisy, Marc Lanctot, Jiri Cermak, Mark H. M. Winands

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review


Simultaneous move games model discrete, multistage interactions where at each stage players simultaneously choose their actions. At each stage, a player does not know what action the other player will take, but otherwise knows the full state of the game. This formalism has been used to express games in general game playing and can also model many discrete approximations of real-world scenarios. In this paper, we describe both novel and existing algorithms that compute strategies for the class of two-player zero-sum simultaneous move games. The algorithms include exact backward induction methods with efficient pruning, as well as Monte Carlo sampling algorithms. We evaluate the algorithms in two different settings: the offline case, where computational resources are abundant and closely approximating the optimal strategy is a priority, and the online search case, where computational resources are limited and acting quickly is necessary. We perform a thorough experimental evaluation on six substantially different games for both settings. For the exact algorithms, the results show that our pruning techniques for backward induction dramatically improve the computation time required by the previous exact algorithms. For the sampling algorithms, the results provide unique insights into their performance and identify favorable settings and domains for different sampling algorithms.
Original languageEnglish
Pages (from-to)1-40
JournalArtificial Intelligence
Publication statusPublished - Aug 2016


  • Simultaneous move games
  • Markov games
  • Backward induction
  • Monte Carlo Tree Search
  • Alpha-beta pruning
  • Double-oracle algorithm
  • Regret matching
  • Counterfactual regret minimization
  • Game playing
  • Nash equilibrium


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