Avoidance Games Are PSPACE-Complete

Authors Valentin Gledel , Nacim Oijid

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Author Details

Valentin Gledel
  • Department of Mathematics and Mathematical Statistics, Umeå University, Sweden
Nacim Oijid
  • Univ. Lyon, Université Lyon 1, LIRIS UMR CNRS 5205, F-69621, Lyon, France


We want to thank Eric Duchêne, Marianne Fortin, Aline Parreau and the anonymous referees for their help in the writing of this article.

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Valentin Gledel and Nacim Oijid. Avoidance Games Are PSPACE-Complete. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 34:1-34:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Avoidance games are games in which two players claim vertices of a hypergraph and try to avoid some structures. These games have been studied since the introduction of the game of SIM in 1968, but only few complexity results have been found out about them. In 2001, Slany proved some partial results on Avoider-Avoider games complexity, and in 2017 Bonnet et al. proved that short Avoider-Enforcer games are Co-W[1]-hard. More recently, in 2022, Miltzow and Stojaković proved that these games are NP-hard. As these games correspond to the misère version of the well-known Maker-Breaker games, introduced in 1963 and proven PSPACE-complete in 1978, one could expect these games to be PSPACE-complete too, but the question has remained open since then. Here, we prove here that both Avoider-Avoider and Avoider-Enforcer conventions are PSPACE-complete. Using the PSPACE-hardness of Avoider-Enforcer, we provide in appendix proofs that some particular Avoider-Enforcer games also are.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Games
  • Avoider-Enforcer
  • Maker-Breaker
  • Complexity
  • Avoider-Avoider
  • PSPACE-complete


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