The maximum Theorem and the existence of Nash equilibrium of (generalized) games without lower semicontinuities

Abstract

In this paper we generalize Berge's Maximum Theorem to the case where the payoff (utility) functions and the feasible action correspondences are not lower semicontinuous. The condition we introduced is called the Feasible Path Transfer Lower Semicontinuity (in short, FPT l.s.c.). By applying our Maximum Theorem to game theory and economics, we are able to prove the existence of equilibrium for the generalized games (the so-called abstract economics) and Nash equilibrium for games where the payoff functions and the feasible strategy correspondences are not lower semicontinuous. Thus the existence theorems given in this paper generalize many existence theorems on Nash equilibrium and equilibrium for the generalized games in the literature.