×

zbMATH — the first resource for mathematics

Credible group stability in many-to-many matching problems. (English) Zbl 1132.91579
Summary: It is known that in two-sided many-to-many matching problems, pairwise-stable matchings may not be immune to group deviations, unlike in many-to-one matching problems [C. Blair, Math. Oper. Res. 13, No. 4, 619–628 (1988; Zbl 0664.90075)]. In this paper, we show that pairwise stability is equivalent to credible group stability when one side has responsive preferences and the other side has categorywise-responsive preferences. A credibly group-stable matching is immune to any “executable” group deviations with an appropriate definition of executability. Under the same preference restriction, we also show the equivalence between the set of pairwise-stable matchings and the set of matchings generated by coalition-proof Nash equilibria of an appropriately defined strategic-form game.

MSC:
91B68 Matching models
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alkan, A., Nonexistence of stable threesome matchings, Math. soc. sci., 16, 207-209, (1988) · Zbl 0651.92025
[2] Alkan, A., On the properties of stable many-to-many matchings under responsive preferences, () · Zbl 0959.91021
[3] R.J. Aumann, Acceptable points in general cooperative N-person games, Contributions to the Theory of Games, vol. IV, Princeton University Press, Princeton, 1959. · Zbl 0085.13005
[4] Bernheim, D.; Peleg, B.; Whinston, M., Coalition-proof Nash equilibria. I. concepts, J. econ. theory, 42, 1-12, (1987) · Zbl 0619.90090
[5] Blair, C., The lattice structure of the set of stable matchings with multiple partners, Math. oper. res., 13, 619-628, (1988) · Zbl 0664.90075
[6] Dutta, B.; Mutuswami, S., Stable networks, J. econ. theory, 76, 322-344, (1997) · Zbl 0893.90043
[7] Dutta, B.; van den Nouweland, A.; Tijs, S., Link formation in cooperative situations, Int. J. game theory, 27, 245-256, (1998) · Zbl 1058.91510
[8] F. Echenique, J. Oviedo, A theory of stability in many-to-many matching markets, Caltech working paper, 2004. · Zbl 1094.91049
[9] Gale, D.; Shapley, L., College admissions and stability of marriage, Amer. math. monthly, 69, 9-15, (1962) · Zbl 0109.24403
[10] J.W. Hatfield, P. Milgrom, Auctions, matching and law of aggregate demand, Stanford University working paper, 2004.
[11] M.O. Jackson, A. van den Nouweland, Strongly stable networks, Games Econ. Behav., doi:10.1016/j.geb.2004.08.004. · Zbl 1099.91011
[12] Kara, T.; Sönmez, T., Implementation of college admission rules, Econ. theory, 9, 197-218, (1997) · Zbl 0872.90006
[13] Kelso, A.S.; Crawford, V.P., Job matching, coalition formation, and Gross substitutes, Econometrica, 50, 1483-1504, (1982) · Zbl 0503.90019
[14] Martínez, R.; Massó, J.; Neme, A.; Oviedo, J., Single agents and the set of many-to-one stable matchings, J. econ. theory, 91, 91-105, (2000) · Zbl 0955.91048
[15] Ray, D., Credible coalitions and the core, Int. J. game theory, 18, 185-187, (1989) · Zbl 0719.90099
[16] Roth, A.E., The evolution of the labor market for medical interns and residents: a case study in game theory, J. polit. economy, 92, 991-1016, (1984)
[17] Roth, A.E., Stability and polarization of interests in job matching, Econometrica, 52, 47-57, (1984) · Zbl 0526.90012
[18] Roth, A.E., The college admissions problem is not equivalent to the marriage problem, J. econ. theory, 36, 277-288, (1985) · Zbl 0594.90002
[19] Roth, A.E., Conflict and coincidence of interest in job matching, Math. oper. res., 10, 379-389, (1985) · Zbl 0586.90107
[20] Roth, A.E., A natural experiment in the organization of entry level labor markets: regional markets for new physicians and surgeons in the UK, Amer. econ. rev., 81, 415-440, (1991)
[21] Roth, A.E.; Peranson, E., The redesign of the matching market for American physicians: some engineering aspects of economic design, Amer. econ. rev., 89, 748-780, (1999)
[22] Roth, A.E.; Sotomayor, M., Two sided matching: A study in game-theoretic modelling and analysis, (1990), Cambridge University Press Cambridge · Zbl 0726.90003
[23] Sönmez, T., Strategy-proofness and essentially single-valued cores, Econometrica, 67, 677-689, (1999) · Zbl 1021.91036
[24] Sotomayor, M., Three remarks on the many-to-many stable matching problem, Math. soc. sci., 38, 55-70, (1999) · Zbl 0951.91045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.