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Matching of like rank and the size of the core in the marriage problem

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  • Holzman, Ron
  • Samet, Dov

Abstract

When men and women are objectively ranked in a marriage problem, say by beauty, then pairing individuals of equal rank is the only stable matching. We generalize this observation by providing bounds on the size of the rank gap between mates in a stable matching in terms of the size of the ranking sets. Using a metric on the set of matchings, we provide bounds on the diameter of the core – the set of stable matchings – in terms of the size of the ranking sets and in terms of the size of the rank gap. We conclude that when the set of rankings is small, so are the core and the rank gap in stable matchings. We construct examples showing that our bounds are essentially tight, and that certain natural variants of the bounds fail to hold.

Suggested Citation

  • Holzman, Ron & Samet, Dov, 2014. "Matching of like rank and the size of the core in the marriage problem," Games and Economic Behavior, Elsevier, vol. 88(C), pages 277-285.
    • Handle: RePEc:eee:gamebe:v:88:y:2014:i:c:p:277-285
    DOI: 10.1016/j.geb.2014.10.003
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    References listed on IDEAS

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    1. James W. Boudreau & Vicki Knoblauch, 2010. "Marriage Matching and Intercorrelation of Preferences," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 12(3), pages 587-602, June.
    2. Clark Simon, 2006. "The Uniqueness of Stable Matchings," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 6(1), pages 1-30, December.
    3. Fuhito Kojima & Parag A. Pathak, 2009. "Incentives and Stability in Large Two-Sided Matching Markets," American Economic Review, American Economic Association, vol. 99(3), pages 608-627, June.
    4. Eeckhout, Jan, 2000. "On the uniqueness of stable marriage matchings," Economics Letters, Elsevier, vol. 69(1), pages 1-8, October.
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    6. Caldarelli, G. & Capocci, A., 2001. "Beauty and distance in the stable marriage problem," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 300(1), pages 325-331.
    7. Elliott Peranson & Alvin E. Roth, 1999. "The Redesign of the Matching Market for American Physicians: Some Engineering Aspects of Economic Design," American Economic Review, American Economic Association, vol. 89(4), pages 748-780, September.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Paula Jaramillo & Cagatay Kayi & Flip Klijn, 2017. "Rank Gaps and the Size of the Core for Roommate Problems," Documentos CEDE 15610, Universidad de los Andes, Facultad de Economía, CEDE.
    2. Mao, Fubing & Ma, Lijia & He, Qiang & Xiao, Gaoxi, 2020. "Match making in complex social networks," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    3. Marcelo Ariel Fernandez & Kirill Rudov & Leeat Yariv, 2022. "Centralized Matching with Incomplete Information," American Economic Review: Insights, American Economic Association, vol. 4(1), pages 18-33, March.
    4. Karpov, Alexander, 2019. "A necessary and sufficient condition for uniqueness consistency in the stable marriage matching problem," Economics Letters, Elsevier, vol. 178(C), pages 63-65.
    5. Kristian Koerselman, 2020. "Why Finnish polytechnics reject top applicants," Education Economics, Taylor & Francis Journals, vol. 28(5), pages 491-507, September.
    6. Gregory Z. Gutin & Philip R. Neary & Anders Yeo, 2021. "Unique Stable Matchings," Papers 2106.12977, arXiv.org, revised Jul 2023.
    7. Flip Klijn & Markus Walzl & Christopher Kah, 2021. "Almost mutually best in matching markets: rank gaps and size of the core," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 57(4), pages 797-816, November.
    8. Ortega, Josué, 2018. "Social integration in two-sided matching markets," Journal of Mathematical Economics, Elsevier, vol. 78(C), pages 119-126.
    9. Paula Jaramillo & Çaǧatay Kayı & Flip Klijn, 2019. "The core of roommate problems: size and rank-fairness within matched pairs," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(1), pages 157-179, March.
    10. Rheingans-Yoo, Ross, 2024. "Large random matching markets with localized preference structures can exhibit large cores," Games and Economic Behavior, Elsevier, vol. 144(C), pages 71-83.
    11. Peng, Zixuan & Shan, Wenxuan & Guan, Feng & Yu, Bin, 2016. "Stable vessel-cargo matching in dry bulk shipping market with price game mechanism," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 95(C), pages 76-94.
    12. Christopher Kah & Flip Klijn & Markus Walzl, 2019. "Almost Mutually Best in Matching Markets: Rank-Fairness and Size of the Core," Working Papers 1115, Barcelona School of Economics.
    13. Rouzbeh Ghouchani & Szilvia Pápai, 2022. "Preference aggregation for couples," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 59(4), pages 889-923, November.
    14. Gutin, Gregory Z. & Neary, Philip R. & Yeo, Anders, 2023. "Unique stable matchings," Games and Economic Behavior, Elsevier, vol. 141(C), pages 529-547.
    15. Hans Gersbach & Hans Haller, 2015. "Matching on Bipartite Graphs," CESifo Working Paper Series 5575, CESifo.

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    More about this item

    Keywords

    Marriage problem; Stable matching; Core; Correlated preferences; Assortative matching;
    All these keywords.

    JEL classification:

    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory

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