Equivalences between two matching models: Stability

Abstract

We study the equivalences between two matching models, wherethe agents in one side of the market, the workers, have responsive preferenceson the set of agents of the other side, the firms. We modify the firms? preferenceson subsets of workers and define a function between the set of manyto-manymatchings and the set of related many-to-one matchings. We provethat this function restricted to the set of stable matchings is bijective and thatpreserves the stability of the corresponding matchings in both models. Usingthis function, we prove that for the many-to-many problem with substitutablepreferences for the firms and responsive preferences for the workers, the set ofstable matchings is non-empty and has a lattice structure.