Subgame perfect implementation of men-optimal matchings

In this study, we explore monotonicities and implementability of di erent matching rules. We nd a self-monotonicity of the stable rule and an h-monotonicity for the men-optimal rule, which does not satisfy Maskin-monotonicity. We then o er a sequential matching mechanism that implements the men-optimal rule in subgame perfect equilibrium, when there is no other matching that weakly Pareto-dominates the men-optimal matching for men. In our mechanism, women propose to men in an arbitrary hierarchy order, and each man either accepts or rejects the proposals he receives, where accepting means permanent matching with the proposing woman.