Zbigniew Świtalski

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We define, for some variant of a many-to-many market model of Gale-Shapley type, a concept of generalized competitive equilibrium and show that, under suitable conditions, stable matchings in such a model can be represented as competitive equilibria allocations (and vice versa). Our results are far-reaching generalizations of the “discrete supply and demand lemma” of Azevedo, Leshno (2011) for the college admissions market.
Using the results of Alkan, Gale (2003), we also prove a theorem on existence of generalized equilibria in our model.


stable matching, Gale-Shapley theory, many-to-many model, competitive equilibrium, discrete market model, contract theory


Aizerman M., Aleskerov F., (1995), Theory of Choice, North-Holland, Amsterdam.

Aleskerov F., Monjardet B., (2002), Utility Maximization, Choice and Preference, Springer, Berlin.

Alkan A., Gale D., (2003), Stable Schedule Matching under Revealed Preference, Journal of Economic Theory, 112, 289–306.

Azevedo E. M., Leshno J. D., (2011), A Supply and Demand Framework for Two-Sided Matching Markets, working paper, www.aeaweb.org (last access: 25.11.2015).

Biro P., Kiselgof S., (2013), College Admissions with Stable Score-Limits, discussion paper, Institute of Economics, Research Centre for Economic and Regional Studies, Hungarian Academy of Sciences, Budapest.

Camina E., (2006), A Generalized Assignment Game, Mathematical Social Sciences, 52, 152–161.

Chen N., Deng X., Ghosh A., (2014), Competitive Equilibria in Matching Markets with Budgets, arxiv.org.

Crawford V. P., Knoer E. M., (1981), Job Matching with Heterogeneous Firms and Workers, Econometrica, 49, 437–450.

Danilov V., Koshevoy G., (2005), Mathematics of Plott Choice Functions, Mathematical Social Sciences, 49, 245–272.

Echenique F., (2007), Counting Combinatorial Choice Rules, Games and Economic Behavior, 58, 231–245.

Echenique F., Oviedo J., (2006), A Theory of Stability in Many-to-Many Matching Markets, Theoretical Economics, 1, 233–273.

Gale D., Shapley L. S., (1962), College Admissions and the Stability of Marriage, American Mathematical Monthly, 69, 9–15.

Hatfield J. W., Milgrom P. L., (2005), Matching with Contracts, American Economic Review, 95, 913–935.

Hatfield J. W., Kominers S. D., Nichifor A., Ostrovsky M., Westkamp A., (2013), Stability and Competitive Equilibrium in Trading Networks, Journal of Political Economy, 121 (5), 966–1005.

Klaus B., Walzl M., (2009), Stable Many-to-Many Matchings with Contracts, Journal of Mathematical Economics, 45, 422–434.

Kominers S. D., (2012), On the Correspondence of Contracts to Salaries in (Many-to-Many) Matching, Games and Economic Behavior, 75, 984–989.

Manlove D. F., (2002), The Structure of Stable Marriage with Indifference, Discrete Applied Mathematics, 122, 167–181.

Mishra D., Talman D., (2010), Characterization of the Walrasian Equilibria of the Assignment Model, Journal of Mathematical Economics, 46, 6–20.

Moore J. C., (2007), General Equilibrium and Welfare Economics, Springer, Berlin.

Roth A. E., Sotomayor M. A., (1992), Two-Sided Matching. A Study in Game-Theoretic Modeling and Analysis, Cambridge University Press.

Sotomayor M., (2007), Connecting the Cooperative and Competitive Structures of the Multiple Partners Assignment Game, Journal of Economic Theory, 134, 155–174.

Shapley L. S., Scarf H., (1974), On Cores and Indivisibility, Journal of Mathematical Economics, 1, 23–37.

Shapley L. S., Shubik M., (1971/72), The Assignment Game I: The Core, International Journal of Game Theory, 1, 111–130.

Shoham Y., Leyton-Brown K., (2009), Multiagent Systems – Algorithmic, Game-Theoretic and Logical Foundations, Cambridge University Press.

Świtalski Z., (2008), Stability and Equilibria in the Matching Models, Scientific Research of the Institute of Mathematics and Computer Science, Częstochowa University of Technology, 2 (7), 77–85.

Świtalski Z., (2010), Equilibria in Markets with Two-Sided Preferences (in Polish), Przegląd Statystyczny, 4 (57), 54–69.

Świtalski Z., (2015), Some Properties of Competitive Equilibria and Stable Matchings in a Gale-Shapley Market Model, Studia Ekonomiczne, Zeszyty Naukowe Uniwersytetu Ekonomicznego w Katowicach, 248, 222–232.

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