On the existence of limit admissible equilibria in discontinuous games

We consider the existence of limit admissible equilibria, i.e. Nash equilibria in which each player assigns zero probability to the interior of the set of his weakly dominated strategies, in (possibly) discontinuous games. We show that standard sufficient conditions for the existence of Nash equilibrium, such as better-reply security, fail to imply the existence of limit admissible equilibria. We then modify better-reply security to obtain a new condition, admissible security, and show that admissible security is sufficient for the existence of limit admissible equilibria. This result implies the existence of limit admissible equilibria in a Bertrand-Edgeworth competition setting with convex costs analogous to that of Maskin (1986). (C) 2018 Elsevier B.V. All rights reserved.