On Equilibrium Refinement for Discontinuous Games

In moving from finite-action to infinite-action games, standard refinements of the Nash equilibrium concept cease to satisfy certain “natural” properties. For instance, perfect equilibria in compact, continuous games need not be admissible. This paper highlights additional properties of two standard refinement specifications that are not inherited by supersets of the set of finite games. The analysis reveals the following about the behavior of perfectness and strategic stability within a class of (possibly) discontinuous games: (1) Equilibria that assign positive probability to the interior of the set of strategies weakly dominated for some player can be chosen; (2) nonadmissible equilibria need not be ruled out when they are weakly dominated by admissible perfect equilibria; and (3) nonadmissible equilibria may be selected when admissible equilibria are ruled out.