Single agents and the set of many-to-one stable matching

Abstract

Some properties of the set of many-to-one stable matchings for firms that have responsive preferences and quotas are not necessarily true when firms' preferences are substitutable. In particular, we provide examples in which firms have substitutable preferences but firms and workers may be "single" in one stable matching and matched in another one. We identify a set of axioms on firms' preferences guaranteeing that the set of unmatched agents is the same under every stable matching. We also propose a weaker condition than responsiveness, called separability with quotas or q-separability, that together with substitutability implies this set of axioms.