Nash implementing non-monotonic social choice rules by awards

Summary.

By a slight generalization of the definition of implementation (called implementation by awards), Maskin monotonicity is no more needed for Nash implementation. In fact, a weaker condition, to which we refer as almost monotonicity is both necessary and sufficient for social choice correspondences to be Nash implementable by awards. Hence our framework paves the way to the Nash implementation of social choice rules which otherwise fail to be Nash implementable. In particular, the Pareto social choice rule, the majority rule and the strong core are almost monotonic (hence Nash implementable by awards) while they are not Maskin monotonic (hence fail to be Nash implementable in the standard framework).