On the Computational Complexity of Decision Problems about Multi-Player Nash Equilibria

We study the computational complexity of decision problems about Nash equilibria in $m$-player games. Several such problems have recently been shown to be computationally equivalent to the decision problem for the existential theory of the reals, or stated in terms of complexity classes, $\exists\mathbb{R}$-complete, when $m\geq 3$. We show that, unless they turn into trivial problems, they are $\exists\mathbb{R}$-hard even for 3-player zero-sum games. We also obtain new results about several other decision problems. We show that when $m\geq 3$ the problems of deciding if a game has a Pareto optimal Nash equilibrium or deciding if a game has a strong Nash equilibrium are $\exists\mathbb{R}$-complete. The latter result rectifies a previous claim of NP-completeness in the literature. We show that deciding if a game has an irrational valued Nash equilibrium is $\exists\mathbb{R}$-hard, answering a question of Bil`o and Mavronicolas, and address also the computational complexity of deciding if a game has a rational valued Nash equilibrium. These results also hold for 3-player zero-sum games. Our proof methodology applies to corresponding decision problems about symmetric Nash equilibria in symmetric games as well, and in particular our new results carry over to the symmetric setting. Finally we show that deciding whether a symmetric $m$-player games has a non-symmetric Nash equilibrium is $\exists\mathbb{R}$-complete when $m\geq 3$, answering a question of Garg, Mehta, Vazirani, and Yazdanbod.