Consider a social choice correspondence as a mapping from preference profiles to lotteries over some finite set of alternatives. A virtually implementable social choice function in Nash equilibrium is defined, under mild domain restrictions it is shown that in societies with at least three individuals all social choice correspondences are virtually implementable in Nash equilibrium. This contrasts with Maskin's classic characterization, which requires monotonicity as a necessary condition for exact implementation in Nash equilibrium. The two person case is considered seperately. While not all two-person social choice functions are virtually implementable, our necessary and sufficient condition is simple, which contrasts with the complex necessary and sufficient conditions for exact implementation.

The main result of this paper is to characterize the class of two-person social choice correspondences which are Nash-implementable. The characterization result is used to formulate domain restrictions which allow the construction of non-dictatorial and Pareto-efficient social choice correspondences which are Nash-implementable.