Abstract
This paper considers a decentralized process in many-to-many matching problems. We show that if agents on one side of the market have substitutable preferences and those on the other side have responsive preferences, then, from an arbitrary matching, there exists a finite path of matchings such that each matching on the path is formed by satisfying a blocking individual or a blocking pair for the previous matching, and the final matching is pairwise-stable. This implies that an associated stochastic process converges to a pairwise-stable matching in finite time with probability one, if each blocking individual or pair is satisfied with a positive probability at each period along the process.
Similar content being viewed by others
References
Alkan A (1999) On the properties of stable many-to-many matchings under responsive preferences. In: Alkan A, Aliprantis CD, Yannelis NC (eds) Current trends in economics: theory and applications. Springer, Berlin Heidelberg New York
Alkan A (2001) On preferences over subsets and the lattice structure of stable matchings. Rev Econ Design 6:99–111
Alkan A (2002) A class of multipartner matching models with a strong lattice structure. Econ Theory 19:737–746
Blair C (1988) The lattice structure of the set of stable matchings with multiple partners. Math Oper Res 13:619–628
Chung K-S (2000) On the existence of stable roommate matchings. Games Econ Behav 33:206–230
Diamantoudi E, Miyagawa E, Xue L (2004) Random paths to stability in the roommate problem. Games Econ Behav 48:18–28
Echenique F, Oviedo J (2006) A theory of stability in many-to-many matching markets. Theor Econ 1: 233–273
Gale D, Shapley L (1962) College admissions and stability of marriage. Am Math Monthly 69:9–15
Hatfield J, Milgrom P (2005) Matching with contracts. Am Econ Rev 95:913–935
Jackson MO, van den Nouweland A (2005) Strongly stable networks. Games Econ Behav 51: 420–444
Jackson MO, Wolinsky A (1996) A strategic model of social and economic networks. J Econ Theory 71:44–74
Kelso AS, Crawford VP (1982) Job matching, coalition formation, and gross substitutes. Econometrica 50:1483–1504
Klaus B, Klijn F (2006) Paths to stability for matching markets with couples. Games Econ Behav (forthcoming)
Knuth DE (1976) Marriages stables. Les Presse de l’Université de Montréal, Montréal
Konishi H, Ünver MU (2006) Credible group-stability in many-to-many matching problems. J Econ Theory 129:57–80
Martínez R, Masso J, Neme A, Oviedo J (2004) An algorithm to compute the full set of many-to-many stable matchings. Math Soc Sci 47:187–210
McVitie DG, Wilson LB (1971) The stable matching problem. Commun ACM 14:486–493
Pápai S (2004) Random paths to stability in Hedonic coalition formation. University of Notre Dame working paper
Roth AE (1984) Stability and polarization of interests in job matching. Econometrica 52:47–57
Roth AE (1985) The college admissions problem is not equivalent to the marriage problem. J Econ Theory 36:277–288
Roth AE (1991) A natural experiment in the organization of entry level labor markets: regional markets for new physicians and surgeons in the UK. Am Econ Rev 81:415–440
Roth AE, Sotomayor M (1990) Two-sided matching: a study in game-theoretic modeling and analysis. Cambridge University Press, Cambridge
Roth AE, Vande Vate JH (1990) Random paths to stability in two-sided matching. Econometrica 58:1475–1480
Sotomayor M (1999) Three remarks on the many-to-many stable matching problem. Math Soc Sci 38:55–70
Sotomayor M (2004) Implementation in the many-to-many matching market. Games Econ Behav 46:199–212
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kojima, F., Ünver, M.U. Random paths to pairwise stability in many-to-many matching problems: a study on market equilibration. Int J Game Theory 36, 473–488 (2008). https://doi.org/10.1007/s00182-006-0037-2
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00182-006-0037-2