Semidefinite Programming and Nash Equilibria in Bimatrix Games

We explore the power of semidefinite programming (SDP) for finding additive -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 (generalized) zero-sum game, our SDP is guaranteed to return a rank-1 solution. 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 propose two algorithms based on iterative linearization of smooth nonconvex objective functions which are designed to produce rank-1 solutions. Empirically, we show that these algorithms often recover solutions of rank at most two and close to zero. 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 that by using the Lasserre/sum of squares hierarchy, we can get an arbitrarily close spectrahedral outer approximation to the convex hull of Nash equilibria, and that the SDP proposed in this paper dominates the first level of the sum of squares hierarchy.