Generalized Nash equilibrium problems for lower semi-continuous strategy maps

Abstract

We consider a class of generalized Ky Fan inequalities (quasi-variational inequalities) in which the involved multi-valued mapping is lower semi-continuous. We present a relaxed version of the generalized Nash equilibrium problem involving strategy maps, which are only lower semi-continuous. This relaxed version may have no exact Nash equilibrium, but we prove that it has an e-Nash equilibrium for every e > 0. Easy examples of such problems show no existence of exact solutions, but existence of e-solutions for every e > 0. We give positive answers to two questions (in the compact case) raised in a recent paper of Cubiotti and Yao.