Existence of Nash Equilibria in Games with Arbitrary Strategy Spaces and Preferences: A Full Characterization∗

This paper provides a complete solution to the existence of equilibria in games with any number of players that may be finite, infinite, or even uncountable; arbitrary strategy spaces that may be discrete, continuum, non-compact or non-convex; payoffs (resp. preferences) that may be discontinuous or do not have any form of quasi-concavity (resp. nontotal, nontransitive, discontinuous, nonconvex, or nonmonotonic). We establish a single condition, recursive diagonal transfer continuity for aggregate payoffs or recursive weak transfer quasi-continuity for individuals’ preferences, which is necessary and further, under compactness of strategy space, sufficient for the existence of Nash equilibria. Also, we introduce a non-fixed point theoretic proof of equilibrium existence.