  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.
