The Existence of Maximal Elements and Equilibria in the Absence of Transitivity

Ted Bergstrom, University of California, Santa Barbara

Abstract

This paper generalizes results of Ky Fan and Hugo Sonnenschein on the existence of maximal elements for non-transitive binary relations. It proves a generalization of Nash's theorem on the existence of non-cooperative equilibrium. It also shows that existence of competitive equilibrium can be proved as a consequence of the existence of a maximal element for an appropriately chosen binary relation.