Existence of Equilibria in Games with Arbitrary Strategy Spaces and Preferences∗

This paper considers the existence of Nash 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 guarantees the existence of Nash equilibria for any game. Moreover, the condition is also necessary for the existence of equilibrium in games with arbitrary strategy spaces and preferences.