Settling the complexity of computing approximate two-player Nash equilibria

TL;DR

This paper establishes quasi-polynomial lower bounds for computing approximate Nash equilibria in two-player games under the Exponential Time Hypothesis, and proves exponential query complexity for n-player games, using novel PCP techniques.

Contribution

It introduces the first PCP-based reduction inside PPAD and proves new hardness results for multi-player Nash equilibria.

Findings

Computing an ε-approximate Nash equilibrium in two-player games requires quasi-polynomial time under ETH.

Finding an ε-approximate Nash equilibrium in n-player games requires exponential oracle queries.

New hardness results for Nash equilibria in multi-player games with respect to WeakNash relaxation.

Abstract

We prove that there exists a constant $\epsilon>0$ such that, assuming the Exponential Time Hypothesis for PPAD, computing an $\epsilon$-approximate Nash equilibrium in a two-player (nXn) game requires quasi-polynomial time, $n^{\log^{1-o(1)} n}$. This matches (up to the o(1) term) the algorithm of Lipton, Markakis, and Mehta [LMM03]. Our proof relies on a variety of techniques from the study of probabilistically checkable proofs (PCP); this is the first time that such ideas are used for a reduction between problems inside PPAD. En route, we also prove new hardness results for computing Nash equilibria in games with many players. In particular, we show that computing an $\epsilon$-approximate Nash equilibrium in a game with n players requires $2^{\Omega(n)}$ oracle queries to the payoff tensors. This resolves an open problem posed by Hart and Nisan [HN13], Babichenko [Bab14], and Chen et al. [CCT15]. In fact, our results for n-player games are stronger: they hold with respect to the $(\epsilon,\delta)$-WeakNash relaxation recently introduced by Babichenko et al. [BPR16].