Algebraic Degrees of Generalized Nash Equilibrium Problems

TL;DR

This paper investigates the algebraic degrees of solutions to polynomial-based generalized Nash equilibrium problems, providing formulas and conditions that determine the number of complex solutions under generic assumptions.

Contribution

It introduces formulas for algebraic degrees of GNEPs and shows that, generically, all Fritz-John points are KKT points with finitely many complex solutions.

Findings

Finitely many complex Fritz-John points under generic conditions

All Fritz-John points are KKT points in generic cases

Provides algebraic degree formulas for GNEPs

Abstract

This paper studies algebraic degree of generalized Nash equilibrium problems (GNEPs) given by polynomials. Their generalized Nash equilibria (GNEs), as well as their KKT or Fritz-John points, are algebraic functions in the coefficients of defining polynomials. We study the degrees of these algebraic functions, which also counts the numbers of complex KKT or Fritz-John points. Under some genericity assumptions, we show that a GNEP has only finitely many complex Fritz-John points and every Fritz-John point is a KKT point. We also give formulae for algebraic degrees of GNEPs, which count the numbers of complex Fritz-John points for generic cases.