Many-to-one matchings without substitutability

We study the existence of stable matchings in the many-to-one college admission problem when there are no restrictions on college preferences. We show that the existence of a stable allocation is strongly tied to students having “harmonious preferences” over their sets of acceptable college choices. In other words, without any assumption on college preferences, a stable matching exists for any college admission problem if and only if there is no subset of students with misaligned college rankings.