Sum-of-squares meets Nash: lower bounds for finding any equilibrium

Abstract

Computing Nash equilibrium (NE) in two-player game is a central question in algorithmic game theory. The main motivation of this work is to understand the power of sum-of-squares method in computing equilibria, both exact and approximate. Previous works in this context have focused on hardness of approximating “best” equilibria with respect to some natural quality measure on equilibria such as social welfare. Such results, however, do not directly relate to the complexity of the problem of finding any equilibrium. In this work, we propose a framework of roundings for the sum-of-squares algorithm (and convex relaxations in general) applicable to finding approximate/exact equilbria in two player bimatrix games. Specifically, we define the notion of oblivious roundings with verification oracle (OV). These are algorithms that can access a solution to the degree d SoS relaxation to construct a list of candidate (partial) solutions and invoke a verification oracle to check if a candidate in the list gives an (exact or approximate) equilibrium. This framework captures most known approximation algorithms in combinatorial optimization including the celebrated semi-definite programming based algorithms for Max-Cut, Constraint-Satisfaction Problems, and the recent works on SoS relaxations for Unique Games/Small-Set Expansion, Best Separable State, and many problems in unsupervised machine learning. Our main results are strong unconditional lower bounds in this framework. Specifically, we show that for є = Θ(1/poly(n)), there’s no algorithm that uses a o(n)-degree SoS relaxation to construct a 2o(n)-size list of candidates and obtain an є-approximate NE. For some constant є, we show a similar result for degree o(log(n)) SoS relaxation and list size no(log(n)). Our results can be seen as an unconditional confirmation, in our restricted algorithmic framework, of the recent Exponential Time Hypothesis for PPAD. Our proof strategy involves constructing a family of games that all share a common sum-of-squares solution but every (approximate) equilibrium of any game is far from every equilibrium of any other game in the family (in either player’s strategy). Along the way, we strengthen the classical unconditional lower bound against enumerative algorithms for finding approximate equilibria due to Daskalakis-Papadimitriou and the classical hardness of computing equilibria due to Gilbow-Zemel.