A mechanism implementing the stable rule in marriage problems

We consider a stable mechanism where each agent announces his/her own preferences and a stable matching is selected for the announced preference profile. We show that the mechanism implements the stable rule that selects all of the stable matchings in a strong equilibrium. In other words, when we select any stable matching to an announced preference profile, in fact the matching is stable for the true preference profile in environments where agentsʹ behaviors are described by a strong Nash equilibrium.