Self-implementation of social choice correspondences in Nash equilibrium

A social choice correspondence is Nash self-implementable if it can be implemented in Nash equilibrium by a social choice function that selects from it as the game form. We provide a complete characterization of all unanimous and anonymous Nash self-implementable social choice correspondences when there are two agents or two alternatives. For the case of three agents and three alternatives, only the top correspondence is Nash self-implementable. In all other cases, every Nash self-implementable social choice correspondence contains the top correspondence and is contained in the Pareto correspondence. In particular, when the number of alternatives is at least four, every social choice correspondence containing the top correspondence plus the intersection of the Pareto correspondence with a fixed set of alternatives, is self-implementable.