# American Institute of Mathematical Sciences

• Previous Article
Critical transitions and Early Warning Signals in repeated Cooperation Games
• JDG Home
• This Issue
• Next Article
Imperfectly competitive markets, trade unions and inflation: Do imperfectly competitive markets transmit more inflation than perfectly competitive ones? A theoretical appraisal
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 and 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.

show all references

##### 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.
 [1] Azam Chaudhry, Rehana Naz. Closed-form solutions for the Lucas-Uzawa growth model with logarithmic utility preferences via the partial Hamiltonian approach. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 643-654. doi: 10.3934/dcdss.2018039 [2] Kariane Calta, Thomas A. Schmidt. Infinitely many lattice surfaces with special pseudo-Anosov maps. Journal of Modern Dynamics, 2013, 7 (2) : 239-254. doi: 10.3934/jmd.2013.7.239 [3] Guihong Fan, Yijun Lou, Horst R. Thieme, Jianhong Wu. Stability and persistence in ODE models for populations with many stages. Mathematical Biosciences & Engineering, 2015, 12 (4) : 661-686. doi: 10.3934/mbe.2015.12.661 [4] Pablo Neme, Jorge Oviedo. A note on the lattice structure for matching markets via linear programming. Journal of Dynamics and Games, 2021, 8 (1) : 61-67. doi: 10.3934/jdg.2021001 [5] Lea Ellwardt, Penélope Hernández, Guillem Martínez-Cánovas, Manuel Muñoz-Herrera. Conflict and segregation in networks: An experiment on the interplay between individual preferences and social influence. Journal of Dynamics and Games, 2016, 3 (2) : 191-216. doi: 10.3934/jdg.2016010 [6] Chi Zhou, Wansheng Tang, Ruiqing Zhao. Optimal consumption with reference-dependent preferences in on-the-job search and savings. Journal of Industrial and Management Optimization, 2017, 13 (1) : 505-529. doi: 10.3934/jimo.2016029 [7] Cui-Ping Cheng, Wan-Tong Li, Zhi-Cheng Wang. Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 559-575. doi: 10.3934/dcdsb.2010.13.559 [8] Fabio Bagagiolo, Rosario Maggistro, Raffaele Pesenti. Origin-to-destination network flow with path preferences and velocity controls: A mean field game-like approach. Journal of Dynamics and Games, 2021, 8 (4) : 359-380. doi: 10.3934/jdg.2021007 [9] Yang Yang, Guanxin Yao. Fresh agricultural products supply chain coordination considering consumers' dual preferences under carbon cap-and-trade mechanism. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022032 [10] Motoko Qiu Kawakita. Certain sextics with many rational points. Advances in Mathematics of Communications, 2017, 11 (2) : 289-292. doi: 10.3934/amc.2017020 [11] D. Alderson, H. Chang, M. Roughan, S. Uhlig, W. Willinger. The many facets of internet topology and traffic. Networks and Heterogeneous Media, 2006, 1 (4) : 569-600. doi: 10.3934/nhm.2006.1.569 [12] Rodica Toader. Scattering in domains with many small obstacles. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 321-338. doi: 10.3934/dcds.1998.4.321 [13] Marco Di Francesco, Simone Fagioli, Massimiliano D. Rosini. Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic. Mathematical Biosciences & Engineering, 2017, 14 (1) : 127-141. doi: 10.3934/mbe.2017009 [14] Eleonora Catsigeras, Marcelo Cerminara, Heber Enrich. Simultaneous continuation of infinitely many sinks at homoclinic bifurcations. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 693-736. doi: 10.3934/dcds.2011.29.693 [15] Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many solutions for a perturbed Schrödinger equation. Conference Publications, 2015, 2015 (special) : 94-102. doi: 10.3934/proc.2015.0094 [16] Andrzej Szulkin, Shoyeb Waliullah. Infinitely many solutions for some singular elliptic problems. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 321-333. doi: 10.3934/dcds.2013.33.321 [17] Jozef Bobok, Martin Soukenka. On piecewise affine interval maps with countably many laps. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 753-762. doi: 10.3934/dcds.2011.31.753 [18] Vilmos Komornik, Anna Chiara Lai, Paola Loreti. Simultaneous observability of infinitely many strings and beams. Networks and Heterogeneous Media, 2020, 15 (4) : 633-652. doi: 10.3934/nhm.2020017 [19] Jon Chaika, Bryna Kra. A prime system with many self-joinings. Journal of Modern Dynamics, 2021, 17: 213-265. doi: 10.3934/jmd.2021007 [20] Nataliya Goncharuk, Yury Kudryashov. Families of vector fields with many numerical invariants. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 239-259. doi: 10.3934/dcds.2021114

Impact Factor: