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

Abstract

Abstract. 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. All our results extend to k-Nash, for any constant k ≥ 3.

The art gallery problem is ∃ ℝ-complete

“…graph has a straight line drawing with a given number of edge crossings [10], decision problems related to Nash equilibria [24], positive semidefinite matrix factorization [49], and nonnegative matrix factorization [48]. We refer the reader to the lecture notes by Matoušek [33] and surveys by Schaefer [43] and Cardinal [15] for more information on the complexity class ∃R.…”

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

“…It consists of the problems whose resolution on an instance reduces to the computation of a fixed point of some function F that can be expressed by the operations {+, * , −, /, max, min} with rational constants and functions and computed in time polynomial in the size of ; this extends PPAD, which coincides with the case of linear functions. The second class, called ∃R [340] (sometimes abbreviated ETR for existence of real solutions, see [180]), studies problems that reduce to deciding the emptiness of a general semi-algebraic set, i.e., the set of real solutions of a system of inequalities with polynomials as constraints. These two complexity classes are relevant in Section 4.…”

“…The problem of computing a Nash equilibrium for two players is PPAD-complete [103] (see also [120]). The intractability for games with three or more players is even more stringent, as many decision problems are ∃R-complete, i.e., as difficult as deciding the emptiness of a general semi-algebraic set; this includes in particular deciding whether a 3-player game has a Nash equilibrium within ∞ -distance r from a given distribution x [340, Corollary 5.5] or the existence of more than one equilibrium or of equilibria with payoff or support conditions [180]. Behind this ∃R-completeness lurks a more daunting fact: Datta's universality theorem [121] asserts that arbitrarily complicated semialgebraic sets can be encoded as sets of Nash equilibria (formally: every real algebraic variety is isomorphic to the set of mixed Nash equilibria of some 3-player game).…”

Inapproximability Results for Approximate Nash Equilibria

“…Conitzer and Sandholm [3] extended the list of NP-complete problems of [2] and furthermore proved inapproximability results for some of them. Recently, Garg et al [4] and Bilo and Mavronicolas [5,6] extended these results to many player games and provided ETR-completeness results for them. Approximate equilibria.…”