Equilibria under deferred acceptance: Dropping strategies, filled positions, and welfare

We study many–to–one matching markets where hospitals have responsive preferences over students. We study the game induced by the student-optimal stable matching mechanism. We assume that students play their weakly dominant strategy of truth-telling. Roth and Sotomayor (1990) showed that equilibrium outcomes can be unstable. We prove that any stable matching is obtained in some equilibrium. We also show that the exhaustive class of dropping strategies does not necessarily generate the full set of equilibrium outcomes. Finally, we find that the ‘rural hospital theorem’ cannot be extended to the set of equilibrium outcomes and that welfare levels are in general unrelated to the set of stable matchings. Two important consequences are that, contrary to one–to–one matching markets, (a) filled positions depend on the equilibrium that is reached and (b) welfare levels are not bounded by the optimal stable matchings (with respect to the true preferences).