A decomposition from a many-to-one matching market with path-independent choice functions to a one-to-one matching market
Pablo Neme, Jorge Oviedo
TL;DR
This work links many-to-one markets with path-independent firm choices to associated one-to-one markets using the Aizerman-Malishevski decomposition. It introduces stability* to capture appropriate stability when firms are represented by multiple copies with differing preferences, and proves an isomorphism between stable matchings in the original market and stable* matchings in the decomposed market. The authors adapt the deferred acceptance algorithm to compute stable* outcomes from either side and show that the stable* set is non-empty; they also obtain an adapted Rural Hospital Theorem via the isomorphism. Overall, the paper provides a principled method to transfer stability concepts and algorithmic procedures across market formats, with implications for resource allocation and matching design in settings with capacities and path-dependent choices.
Abstract
For a many-to-one market where firms are endowed with path-independent choice functions, based on the Aizerman-Malishevski decomposition, we define an associated one-to-one market. Given that the usual notion of stability for a one-to-one market does not fit well for this associated one-to-one market, we introduce a new notion of stability. This notion allows us to establish an isomorphism between the set of stable matchings in the many-to-one market and the matchings in an associated one-to-one market that meet this new stability criterion. Furthermore, we present an adaptation of the well-known deferred acceptance algorithm to compute a matching that satisfies this new notion of stability for the associated one-to-one market. Finally, as a byproduct of our isomorphism, we prove an adapted version of the so-called Rural Hospital Theorem.