Avoidance games are PSPACE-Complete

TL;DR

This paper proves that avoidance games, where players claim vertices to avoid certain structures, are PSPACE-complete, resolving an open question and extending complexity results known for related combinatorial games.

Contribution

The paper establishes that both Avoider-Avoider and Avoider-Enforcer games are PSPACE-complete, filling a gap in the complexity classification of these combinatorial games.

Findings

Avoider-Avoider games are PSPACE-complete.

Avoider-Enforcer games are PSPACE-complete.

Some specific Avoider-Enforcer games are also PSPACE-complete.

Abstract

Avoidance games are games in which two players claim vertices of a hypergraph and try to avoid some structures. These games are studied since the introduction of the game of SIM in 1968, but only few complexity results are known on 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 Stojakovi\'c proved that these games are NP-hard. As these games corresponds to the mis\`ere 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 remained open since then. We prove here that both Avoider-Avoider and Avoider-Enforcer conventions are PSPACE-complete, and as a consequence of it that some particular Avoider-Enforcer games also are.