Semidefinite Programming and Nash Equilibria in Bimatrix Games

TL;DR

This paper introduces a semidefinite programming approach to approximate Nash equilibria in bimatrix games, providing algorithms and theoretical guarantees for solutions close to exact equilibria and addressing related economic problems.

Contribution

The paper develops an SDP relaxation for Nash equilibria, proposes algorithms for rank-1 solutions, and connects the approach to the Lasserre hierarchy, advancing computational methods in game theory.

Findings

SDP relaxation can find approximate Nash equilibria with low rank solutions.

Algorithms often recover solutions of rank at most two with epsilon close to zero.

Rank-2 solutions lead to approximate equilibria with known guarantees.

Abstract

We explore the power of semidefinite programming (SDP) for finding additive $epsilon$-approximate Nash equilibria in bimatrix games. We introduce an SDP relaxation for a quadratic programming formulation of the Nash equilibrium (NE) problem and provide a number of valid inequalities to improve the quality of the relaxation. If a rank-1 solution to this SDP is found, then an exact NE can be recovered. We show that for a strictly competitive game, our SDP is guaranteed to return a rank-1 solution. We propose two algorithms based on iterative linearization of smooth nonconvex objective functions whose global minima by design coincide with rank-1 solutions. Empirically, we demonstrate that these algorithms often recover solutions of rank at most two and $epsilon$ close to zero. Furthermore, we prove that if a rank-2 solution to our SDP is found, then a 5/11-NE can be recovered for any game, or a 1/3-NE for a symmetric game. We then show how our SDP approach can address two (NP-hard) problems of economic interest: finding the maximum welfare achievable under any NE, and testing whether there exists a NE where a particular set of strategies is not played. Finally, we show the connection between our SDP and the first level of the Lasserre/sum of squares hierarchy.