IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2603.23385.html

The Distribution of Envy in Matching Markets

Author

Listed:
  • Josu'e Ortega
  • Gabriel Ziegler
  • R. Pablo Arribillaga
  • Geng Zhao

Abstract

We study the distribution of envy in random matching markets under the Deferred Acceptance (DA) algorithm. Using tools from applied probability, we compute the expected number of proposing agents whom nobody envies and those who envy nobody. We obtain an exact finite-market expression for the former, based on a connection with the coupon collector problem, and asymptotic bounds for the latter. To put these quantities into perspective, we compare them to their counterparts under Random Serial Dictatorship (RSD): while RSD assigns a constant fraction of agents to their top choice, both DA and RSD leave exactly $H_n$ proposing agents unenvied in expectation. Our results show that these clearly unimprovable proposing agents constitute a vanishing fraction of the market.

Suggested Citation

  • Josu'e Ortega & Gabriel Ziegler & R. Pablo Arribillaga & Geng Zhao, 2026. "The Distribution of Envy in Matching Markets," Papers 2603.23385, arXiv.org.
    • Handle: RePEc:arx:papers:2603.23385
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2603.23385
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Tang, Qianfeng & Yu, Jingsheng, 2014. "A new perspective on Kesten's school choice with consent idea," Journal of Economic Theory, Elsevier, vol. 154(C), pages 543-561.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Rodríguez-Álvarez, Carmelo & Romero-Medina, Antonio, 2024. "School choice with transferable student characteristics," Games and Economic Behavior, Elsevier, vol. 143(C), pages 103-124.
    2. Minoru Kitahara & Yasunori Okumura, 2023. "On extensions of partial priorities in school choice," Papers 2305.00641, arXiv.org, revised Oct 2023.
    3. Jiao, Zhenhua & Tian, Guoqiang, 2018. "Two further impossibility results on responsive affirmative action in school choice," Economics Letters, Elsevier, vol. 166(C), pages 60-62.
    4. Zhang, Luosai, 2025. "The properness of weak stability notions," Economics Letters, Elsevier, vol. 247(C).
    5. Imamura, Kenzo & Kawase, Yasushi, 2024. "Efficient matching under general constraints," Games and Economic Behavior, Elsevier, vol. 145(C), pages 197-207.
    6. Troyan, Peter & Delacrétaz, David & Kloosterman, Andrew, 2020. "Essentially stable matchings," Games and Economic Behavior, Elsevier, vol. 120(C), pages 370-390.
    7. Yasushi Kawase & Keisuke Bando, 2021. "Subgame perfect equilibria under the deferred acceptance algorithm," International Journal of Game Theory, Springer;Game Theory Society, vol. 50(2), pages 503-546, June.
    8. Minoru Kitahara & Yasunori Okumura, 2021. "Improving efficiency in school choice under partial priorities," International Journal of Game Theory, Springer;Game Theory Society, vol. 50(4), pages 971-987, December.
    9. Kitahara, Minoru & Okumura, Yasunori, 2024. "Extensions of partial priorities and stability in school choice," Mathematical Social Sciences, Elsevier, vol. 131(C), pages 1-4.
    10. Chao Huang & Qianfeng Tang & Ziwei Wang, 2017. "Epsilon-stability in school choice," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 49(2), pages 277-286, August.
    11. Minoru Kitahara & Yasunori Okumura, 2023. "School Choice with Multiple Priorities," Papers 2308.04780, arXiv.org, revised Oct 2024.
    12. Battal Dogan & Lars Ehlers, 2020. "Blocking pairs versus blocking students: Stability comparisons in school choice," Cahiers de recherche 2020-02, Universite de Montreal, Departement de sciences economiques.
    13. Qianfeng Tang & Yongchao Zhang, 2021. "Weak stability and Pareto efficiency in school choice," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(2), pages 533-552, March.
    14. Dur, Umut & Gitmez, A. Arda & Yılmaz, Özgür, 2019. "School choice under partial fairness," Theoretical Economics, Econometric Society, vol. 14(4), November.
    15. Kang Rong & Qianfeng Tang & Yongchao Zhang, 2024. "The core of school choice problems," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 77(3), pages 783-800, May.
    16. Siwei Chen & Yajing Chen & Chia‐Ling Hsu, 2023. "New axioms for top trading cycles," Bulletin of Economic Research, Wiley Blackwell, vol. 75(4), pages 1064-1077, October.
    17. Raghavan, Madhav, 2020. "Influence in private-goods allocation," Journal of Mathematical Economics, Elsevier, vol. 89(C), pages 14-28.
    18. Kasajima, Yoichi & Toda, Manabu, 2024. "Singles monotonicity and stability in one-to-one matching problems," Games and Economic Behavior, Elsevier, vol. 143(C), pages 269-286.
    19. Siwei Chen & Yajing Chen & Chia-Ling Hsu, 2021. "New axioms for top trading cycles," Papers 2104.09157, arXiv.org, revised Jun 2021.
    20. Somouaoga Bonkoungou, 2021. "Decentralized college admissions under single application," Review of Economic Design, Springer;Society for Economic Design, vol. 25(1), pages 65-91, June.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2603.23385. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.