$exists ℝ$–Complete Decision Problems about (Symmetric) Nash Equilibria in (Symmetric) Multi-Player Games

A decision problem about Nash equilibria is concerned with whether a given game has a Nash equilibrium with certain natural properties. We settle the complexity of such decision problems over multi-player games, establishing that (nearly) all decision problems that were before shown NP-complete over 2-player games in References [5, 12, 18] become ∃ℝ-complete over multi-player games. ∃ℝ [27] is the class capturing the complexity of deciding the Existential Theory of the Reals. Specifically, we present a simple, unifying, parametric polynomial time reduction from the problem of deciding, given a 3-player (symmetric) game, whether there is a (symmetric) Nash equilibrium where all strategies played with non-zero probability come from a given set, which was shown ∃ℝ-complete in Reference [17]. By suitably working on the tuning parameters, our reduction delivers two extensive catalogs of ∃ℝ-complete decision problems in multi-player games. The first addresses Nash equilibria in general games, while the second encompasses symmetric Nash equilibria in symmetric games. These results resolve completely the main open problem from Reference [17] to enlarge the class of ∃ℝ-complete decision problems about (symmetric) Nash equilibria in multi-player (symmetric) games.