A decomposition from a many-to-one matching market with path-independent choice functions to a one-to-one matching market

TL;DR

This paper introduces a method to decompose many-to-one matching markets with path-independent choice functions into an associated one-to-one market, establishing a new stability concept and an adapted algorithm for finding stable matchings.

Contribution

It proposes a novel decomposition approach and a new stability notion for matching markets, along with an adapted deferred acceptance algorithm.

Findings

Established an isomorphism between stable matchings in the original and associated markets.

Introduced a new stability concept suited for the associated one-to-one market.

Proved an adapted Rural Hospital Theorem for the decomposed market.

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.