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July  2018, 5(3): 203-221. doi: 10.3934/jdg.2018013

## Equivalences between two matching models: Stability

 Instituto de Matemática Aplicada San Luis, Universidad Nacional de San Luis and CONICET, Italia 1556. D5700HHW San Luis. República Argentina

* Corresponding author

Received  May 2017 Revised  February 2018 Published  May 2018

We study the equivalences between two matching models, where the agents in one side of the market, the workers, have responsive preferences on the set of agents of the other side, the firms. We modify the firms' preferences on subsets of workers and define a function between the set of many-to-many matchings and the set of related many-to-one matchings. We prove that this function restricted to the set of stable matchings is bijective and that preserves the stability of the corresponding matchings in both models. Using this function, we prove that for the many-to-many problem with substitutable preferences for the firms and responsive preferences for the workers, the set of stable matchings is non-empty and has a lattice structure.

Citation: Paola B. Manasero. Equivalences between two matching models: Stability. Journal of Dynamics & Games, 2018, 5 (3) : 203-221. doi: 10.3934/jdg.2018013
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