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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
References:
 [1] G. Birkhoff, Lattice Theory, 2nd edition, American Mathematical Society, Providence, Rhode Island, 1948. [2] C. Blair, The lattice structure of the set of stable matchings with multiple partners, Mathematics of Operations Research, 13 (1988), 619-628. doi: 10.1287/moor.13.4.619. [3] F. Echenique and J. Oviedo, Core many-to-one matchings by fixed point methods, Journal of Economic Theory, 115 (2004), 358-376. doi: 10.1016/S0022-0531(04)00042-1. [4] D. Gale and L. Shapley, College admissions and stability of marriage, American Mathematical Monthly, 69 (1962), 9-15. doi: 10.1080/00029890.1962.11989827. [5] D. Gale and M. Sotomayor, Some remarks on the stable marriage problem, Discrete Applied Mathematics, 11 (1985), 223-232. doi: 10.1016/0166-218X(85)90074-5. [6] J. W. Hatfield and F. Kojima, Substitutes and stability for matching with contracts, Journal of Economic Theory, 145 (2010), 1704-1723. doi: 10.1016/j.jet.2010.01.007. [7] A. Kelso and V. Crawford, Job matching, coalition formation, and gross substitutes, Econometrica, 50 (1982), 1483-1504. doi: 10.2307/1913392. [8] D. Knuth, Marriages Stables, Les Presses de l'Université de Montréal, Montréal. [9] R. Martinez, J. Massó, A. Neme and J. Oviedo, On the lattice structure of the set of stable matchings for a many-to-one model, Optimization, 50 (2001), 439-457. doi: 10.1080/02331930108844574. [10] A. Roth, The evolution of the labor market for medical interns and residents: A case study in game theory, Journal of Political Economy, 92 (1984), 991-1016. doi: 10.1086/261272. [11] A. Roth, Conflict and coincidence of interest in job matching: Some new results and open questions for medical interns and residents: A Case study in game theory, Mathematics Of Operations Research, 10 (1985), 379-389. doi: 10.1287/moor.10.3.379. [12] A. Roth, The college admissions problem is not equivalent to the marriage problem, Journal of Economic Theory, 36 (1985), 277-288. doi: 10.1016/0022-0531(85)90106-1. [13] A. Roth, On the allocation of residents to rural hospitals: A general property of two-sided matching markets, Econometrica, 54 (1986), 425-427. doi: 10.2307/1913160. [14] A. Roth and M. Sotomayor, The college admissions problem revisited, Econometrica, 57 (1989), 559-570. doi: 10.2307/1911052. [15] A. Roth and M. Sotomayor, Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis, Cambridge University Press, Cambridge, 1990. [16] M. Sotomayor, Three remarks on the many-to-many stable matching problem, Mathematical Social Sciences, 38 (1999), 55-70. doi: 10.1016/S0165-4896(98)00048-1.

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References:
 [1] G. Birkhoff, Lattice Theory, 2nd edition, American Mathematical Society, Providence, Rhode Island, 1948. [2] C. Blair, The lattice structure of the set of stable matchings with multiple partners, Mathematics of Operations Research, 13 (1988), 619-628. doi: 10.1287/moor.13.4.619. [3] F. Echenique and J. Oviedo, Core many-to-one matchings by fixed point methods, Journal of Economic Theory, 115 (2004), 358-376. doi: 10.1016/S0022-0531(04)00042-1. [4] D. Gale and L. Shapley, College admissions and stability of marriage, American Mathematical Monthly, 69 (1962), 9-15. doi: 10.1080/00029890.1962.11989827. [5] D. Gale and M. Sotomayor, Some remarks on the stable marriage problem, Discrete Applied Mathematics, 11 (1985), 223-232. doi: 10.1016/0166-218X(85)90074-5. [6] J. W. Hatfield and F. Kojima, Substitutes and stability for matching with contracts, Journal of Economic Theory, 145 (2010), 1704-1723. doi: 10.1016/j.jet.2010.01.007. [7] A. Kelso and V. Crawford, Job matching, coalition formation, and gross substitutes, Econometrica, 50 (1982), 1483-1504. doi: 10.2307/1913392. [8] D. Knuth, Marriages Stables, Les Presses de l'Université de Montréal, Montréal. [9] R. Martinez, J. Massó, A. Neme and J. Oviedo, On the lattice structure of the set of stable matchings for a many-to-one model, Optimization, 50 (2001), 439-457. doi: 10.1080/02331930108844574. [10] A. Roth, The evolution of the labor market for medical interns and residents: A case study in game theory, Journal of Political Economy, 92 (1984), 991-1016. doi: 10.1086/261272. [11] A. Roth, Conflict and coincidence of interest in job matching: Some new results and open questions for medical interns and residents: A Case study in game theory, Mathematics Of Operations Research, 10 (1985), 379-389. doi: 10.1287/moor.10.3.379. [12] A. Roth, The college admissions problem is not equivalent to the marriage problem, Journal of Economic Theory, 36 (1985), 277-288. doi: 10.1016/0022-0531(85)90106-1. [13] A. Roth, On the allocation of residents to rural hospitals: A general property of two-sided matching markets, Econometrica, 54 (1986), 425-427. doi: 10.2307/1913160. [14] A. Roth and M. Sotomayor, The college admissions problem revisited, Econometrica, 57 (1989), 559-570. doi: 10.2307/1911052. [15] A. Roth and M. Sotomayor, Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis, Cambridge University Press, Cambridge, 1990. [16] M. Sotomayor, Three remarks on the many-to-many stable matching problem, Mathematical Social Sciences, 38 (1999), 55-70. doi: 10.1016/S0165-4896(98)00048-1.
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