Characterizing Stability in Many-to-One Matching with Non-Responsive Couples

TL;DR

This paper characterizes the conditions under which stable matchings exist in many-to-one markets with couples, especially when preferences violate responsiveness, providing comprehensive criteria for stability.

Contribution

It offers new theoretical characterizations of stability in complex matching markets with non-responsive couples, extending previous models with weaker preference assumptions.

Findings

Complete characterization of couples' preferences for stability

Necessary and sufficient conditions on institutional preferences

Conditions guaranteeing the existence of stable matchings

Abstract

We study many-to-one matching problems between institutions and individuals, where each institution may be matched to multiple individuals. The matching market includes couples, who view pairs of institutions as complementary. Institutions' preferences over sets of individuals are assumed to satisfy responsiveness, whereas couples' preferences over pairs of institutions may violate responsiveness. In this setting, we first assume that all institutions share a common preference ordering over individuals, and we establish: (i) a complete characterization of all couples' preference profiles for which a stable matching exists, under the additional assumption that couples violate responsiveness only to ensure co-location at the same institution, and (ii) a necessary and sufficient condition on the common institutional preference such that a stable matching exists when couples may violate responsiveness arbitrarily. Next, we relax the common preference assumption, requiring institutions to share a common ranking only over the members of each couple. Under this weaker assumption, we provide: (i) a complete characterization of all couples' preferences for which a stable matching exists, and (ii) a sufficient condition on individuals' preferences that guarantees the existence of a stable matching.