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

Some properties of the set of many-re-one stable matchings for firms that h ave responsive preferences and quotas are not necessarily true when firms' preferences are substitutable. In particular, we provide examples in which firms have substitutable substitutable preferences but firms and workers ma y be "single" in one stable matching and matched in another one. We identif y a set of axioms on firms' preferences guaranteeing that the set of unmatc hed agents is the same under every stable matching. We also propose a weake r condition than responsiveness, called separability with quotas or q-separ ability, that together with substitutability implies this set of axioms. Jo urnal of Economic Literature Classification Number: J41. (C) 2000 Academic Press.