Nash Equilibrium and Welfare Optimality

If A is a set of social alternatives, a social choice rule (SCR) assigns a subset of A to each potential profile of individuals' preferences over A, where the subset is interpreted as the set of "welfare optima". A game form (or "mechanism") implements the social choice rule if, for any potential profile of preferences, (i) any welfare optimum can arise as Nash equilibrium of the game form (implying, in particular, that a Nash equilibrium exists) and, (ii) all Nash equilibria are welfare optimal.