Fixed Points, Nash Equilibria, and the Existential Theory of the Reals

Abstract

We introduce the complexity class ∃R based on the existential theory of the reals. We show that the definition of ∃R is robust in the sense that even the fragment of the theory expressing solvability of systems of strict polynomial inequalities leads to the same complexity class. Several natural and well-known problems turn out to be complete for ∃R; here we show that the complexity of decision variants of fixed-point problems, including Nash equilibria, are complete for this class, complementing work by Etessami and Yannakakis [13].

ETR-Completeness for Decision Versions of Multi-player (Symmetric) Nash Equilibria

“…All but last have been shown to be NP-complete in case of 2-Nash [10,5], and the last one is shown to be ETR-complete in case of 3-Nash [22]. In this paper, we show ETR-completeness for the first four decision problems for k-Nash, and for third and fourth for symmetric k-Nash.…”

“…To show hardness in case of 3-players, we reduce InBox, which is known to be ETR-complete for 3-Nash [22], to each of MaxPayoff, Subset and Superset, and then from MaxPayoff to NonUnique. Hardness for the k-Nash, k > 3, follows since 3-Nash reduces to k-Nash trivially by introducing dummy players.…”

“…How about the complexity of finding a k-Nash solution that satisfies special properties? Due to the inherent difficulty of dealing with irrational numbers, this problem remained open until 2009, when Schaefer andŠtefankovič [22] formally defined class Existential Theory of Reals (ETR), and showed that checking if a 3-player game has a NE in which every strategy is played with probability at most 0.5 (InBox) is ETRcomplete. ETR is the class of "yes" instances of existentially quantified formulas with bases {+, −, * , ∧, ∨, =, <, >} on real numbers; we note that this class was informally known and used earlier than [22], e.g., see [2].…”

“…As a result of some important works [19,6,3,10,5], 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 anď Stefankovič [22]: that checking whether a 3-player Nash Equilibrium (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 [18]. This yields ETR-completeness for symmetric 3-Nash for the last two problems stated above as well as completeness for the class FIXPa, a variant of FIXP for strong approximation.…”

“…As a result of some important works [19,6,3,10,5], 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 anď Stefankovič [22]: that checking whether a 3-player Nash Equilibrium (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.…”