Existence of pure Nash equilibria in discontinuous and non quasiconcave games

In a recent but well known paper, Reny has proved the existence of Nash equilibria for compact and quasiconcave games, with possibly discontinuous payoff functions. In this paper, we prove that the quasiconcavity assumption in Reny's theorem can be weakened: we introduce a measure allowing to localize the lack of quasiconcavity, which allows to refine the analysis of equilibrium existence (I wish to thank P. J. Reny, two anonymous referees and the associated editor for corrections, suggestions and remarks which led to improvements in the paper).