On a Fixed Point Theorem for General Multivalued Mappings on Finite Sets with Applications in Game Theory

We propose a new fixed point theorem that completely characterizes the existence of fixed points for multivalued maps on finite sets. Our result can be seen as a generalization of Abian's fixed point theorem. In the context of finite games, our result can be used to characterize the existence of a Nash equilibrium in pure strategies and can therefore distinguish pure strategy equilibria from mixed strategy equilibria in the celebrated Nash theorem.