Generalized Nash equilibrium problems for lower semi‐continuous strategy maps

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 ϵ‐Nash equilibrium for every ϵ > 0. Easy examples of such problems show no existence of exact solutions, but existence of ϵ‐solutions for every ϵ > 0. We give positive answers to two questions (in the compact case) raised in a recent paper of Cubiotti and Yao.