Computational Complexity of Multi-player Evolutionarily Stable Strategies
Manon Blanc, Kristoffer Arnsfelt Hansen · Lecture notes in computer science · 2021
In this paper we study the computational complexity of computing an evolutionary stable strategy (ESS) in multi-player symmetric games. For two-player games, deciding existence of an ESS is complete for $$\mathrm {\Sigma }^\mathrm {p}_2$$ , the second level of the polynomial time hierarchy. We show that deciding existence of an ESS of a multi-player game is closely connected to the second level of the real polynomial time hierarchy. Namely, we show that the problem is hard for a complexity class we denote as $$\exists ^\mathrm {D}\cdot \forall \mathbb {R}$$ and is a member of $$\exists \forall \mathbb {R}$$ , where the former class restrict the latter by having the existentially quantified variables be Boolean rather then real-valued. As a special case of our results it follows that deciding whether a given strategy is an ESS is complete for $$\forall \mathbb {R}$$ .