ETR-completeness for decision versions of multi-player (Symmetric) nash equilibria

As a result of some important works [4–6, 10, 15], the complexity of 2-player Nash equilibrium is by now well understood, even when equilibria with special properties are desired and when the game is symmetric. However, for multi-player games, when equilibria with special properties are desired, the only result known is due to Schaefer and Štefankovič [18]: that checking whether a 3-player NE (3-Nash) instance has an equilibrium in a ball of radius half in l∞-norm is ETR-complete, where ETR is the class Existential Theory of Reals. Building on their work, we show that the following decision versions of 3-Nash are also ETR-complete: checking whether (i) there are two or more equilibria, (ii) there exists an equilibrium in which each player gets at least h payoff, where h is a rational number, (iii) a given set of strategies are played with non-zero probability, and (iv) all the played strategies belong to a given set. Next, we give a reduction from 3-Nash to symmetric 3-Nash, hence resolving an open problem of Papadimitriou [14]. This yields ETRcompleteness for symmetric 3-Nash for the last two problems stated above as well as completeness for the class FIXP_a, a variant of FIXP for strong approximation. All our results extend to k-Nash, for any constant k ≥ 3.