Nash implementation on the basis of general priorities

We consider a school choice problem under general priorities with ties. Priorities in practice are usually complex since a school may rank students equally or care about an affirmative action policy. Thus, we do not specify a class of priorities already known, but abstractly treat all priorities such that a stable matching exists for all students' preference profiles. For those priorities, it is unknown whether stable matchings are achievable when students are strategic. We show that a stable correspondence is implementable in Nash equilibria. Then, we focus on the Pareto-frontier of stable matchings, which we call constrained efficient stable. We show that, under a reasonable class of priorities, a constrained efficient stable correspondence is Nash implementable if and only if it satisfies Maskin monotonicity. Finally, we identify a necessary and sufficient condition on priorities under which a constrained efficient stable correspondence is Nash implementable.