Nash Equilibrium Problems of Polynomials

Abstract

This paper studies Nash equilibrium problems that are given by polynomial functions. We formulate efficient polynomial optimization problems for computing Nash equilibria. The Lasserre type Moment-SOS relaxations are used to solve them. Under generic assumptions, the method can find a Nash equilibrium if there is one. Moreover, it can find all Nash equilibria if there are finitely many ones of them. The method can also detect nonexistence if there is no Nash equilibrium.

Convex Generalized Nash Equilibrium Problems and Polynomial Optimization

“…If u is not a GNE, one needs to find an efficient method to obtain a different candidate. Such a method is proposed for solving NEPs [45]. For GNEPs, it is not clear how to generalize the method in [45].…”

“…Such a method is proposed for solving NEPs [45]. For GNEPs, it is not clear how to generalize the method in [45]. When the point u is not a GNE, how can we exclude it and find a better candidate?…”

“…In Section 2.3, a polynomial expression for the ith player's Lagrange multipliers exists if and only if the matrix G i (x) is nonsingular. For classical NEPs of polynomials, the nonsingularity holds generically [43,45]. However, this is often not the case for GNEPs.…”

“…For the classical Nash Equilibrium Problems (NEPs) of polynomials, there exist semidefinite relaxation methods [2,45]. Convex GNEPs can be reformulated as variational inequality (VI) or quasi-variational inequality (QVI) problems [11,18,19,34,48].…”

“…This is because the KKT system of a convex GNEP may still be difficult to solve. The set of GNEs may be nonconvex, even for convex NEPs (see [45]). We refer to [13] for a survey on GNEPs.…”