Implementation in mixed Nash equilibrium

A mechanism implements a social choice correspondence f in mixed Nash equilibrium if, at any preference profile, the set of all (pure and mixed) Nash equilibrium outcomes coincides with the set of f-optimal alternatives for all cardinal representations of the preference profile. Unlike Maskinʼs definition, our definition does not require each optimal alternative to be the outcome of a pure equilibrium. We show that set-monotonicity, a weakening of Maskinʼs monotonicity, is necessary for mixed Nash implementation. With at least three players, set-monotonicity and no-veto power are sufficient. Important correspondences that are not Maskin monotonic can be implemented in mixed Nash equilibrium.