A Generalized KKM Theorem and Its Application to the Existence of Pareto Equilibria for Multi-objective Games

A number of classical problems in non-cooperative game theory are formulated as games with noncommensurable vector payoffs. As the existence of Pareto equilibrium is one of the fundamental problems in non-cooperative game theory, it is necessary to study the existence of Pareto equilibrium for multi-objective games. The main purpose of this paper is to generalize the KKM theorem under the non-convexity setting of topological space. Furthermore, as its applications, a weighted Nash equilibrium existence theorem and a Pareto equilibrium existence theorem for multi-objective games are obtained in general topological spaces without any linear and convexity structure. Our results improve and unify the corresponding results in the recently existing literatures.