IDEAS home Printed from https://ideas.repec.org/a/inm/ormoor/v44y2019i1p122-146.html
   My bibliography  Save this article

On Likely Solutions of the Stable Matching Problem with Unequal Numbers of Men and Women

Author

Listed:
  • Boris Pittel

    (Department of Mathematics, The Ohio State University, Columbus, Ohio 43210)

Abstract

Following up a recent work by Ashlagi, Kanoria, and Leshno, we study a stable matching problem with unequal side sizes, n “men” and N > n “women,” whose preferences for a partner are uniformly random and independent. An asymptotic formula for the expected number of stable matchings is obtained. In particular, for N = n + 1 this number is close to n /( e log n ), in notable contrast with ( n log n )/ e , the formula for the balanced case N = n that we obtained in 1988. We associate with each stable matching ℳ the parameters 𝒲(ℳ) and ℋ(ℳ), which are the total rank of “wives” and the total rank of “husbands,” as ranked by their “spouses” in ℳ. We found the deterministic parameters w ( n , N ) and h ( n , N ) such that the set of scaled pairs (𝒲(ℳ)/ w ( n , N ), ℋ(ℳ)/ h ( n , N )) converges to a single point. In particular, w ( n , n + 1) ∼ n log n , h ( n , n + 1) ∼ n 2 /log n . To compare, for the balanced case n = N we previously found that w ( n , n ) = h ( n , n ) = n 3/2 , and that the pairs of scaled total ranks converged to a hyperbolic arc xy = 1, connecting the rank pairs of two extreme stable matchings, men-optimal and women-optimal. We also show that the expected fraction of persons with more than one stable spouse is vanishingly small if N − n ≫ n .

Suggested Citation

  • Boris Pittel, 2019. "On Likely Solutions of the Stable Matching Problem with Unequal Numbers of Men and Women," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 122-146, February.
    • Handle: RePEc:inm:ormoor:v:44:y:2019:i:1:p:122-146
    DOI: 10.1287/moor.2017.0917
    as

    Download full text from publisher

    File URL: https://doi.org/10.1287/moor.2017.0917
    Download Restriction: no
    File URL: https://libkey.io/10.1287/moor.2017.0917?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Itai Ashlagi & Yash Kanoria & Jacob D. Leshno, 2017. "Unbalanced Random Matching Markets: The Stark Effect of Competition," Journal of Political Economy, University of Chicago Press, vol. 125(1), pages 69-98.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Itai Ashlagi & Mark Braverman & Amin Saberi & Clayton Thomas & Geng Zhao, 2020. "Tiered Random Matching Markets: Rank is Proportional to Popularity," Papers 2009.05124, arXiv.org, revised Jan 2021.
    2. Linda Cai & Clayton Thomas, 2019. "Representing All Stable Matchings by Walking a Maximal Chain," Papers 1910.04401, arXiv.org.
    3. Ashlagi, Itai & Nikzad, Afshin, 2020. "What matters in school choice tie-breaking? How competition guides design," Journal of Economic Theory, Elsevier, vol. 190(C).
    4. Itai Ashlagi & Mark Braverman & Geng Zhao, 2023. "Welfare Distribution in Two-sided Random Matching Markets," Papers 2302.08599, arXiv.org.
    5. Mohammad Akbarpour & Yeganeh Alimohammadi & Shengwu Li & Amin Saberi, 2021. "The Value of Excess Supply in Spatial Matching Markets," Papers 2104.03219, arXiv.org.

    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. Michael Bates & Michael Dinerstein & Andrew C. Johnston & Isaac Sorkin, 2022. "Teacher Labor Market Equilibrium and Student Achievement," CESifo Working Paper Series 9551, CESifo.
    2. Michael Greinecker & Christopher Kah, 2018. "Pairwise stable matching in large economies," Graz Economics Papers 2018-01, University of Graz, Department of Economics.
    3. Saraiva, Gustavo, 2021. "An improved bound to manipulation in large stable matches," Games and Economic Behavior, Elsevier, vol. 129(C), pages 55-77.
    4. Ortega, Josué & Klein, Thilo, 2022. "Improving Efficiency and Equality in School Choice," QBS Working Paper Series 2022/02, Queen's University Belfast, Queen's Business School.
    5. Ortega, Josué, 2018. "Social integration in two-sided matching markets," Journal of Mathematical Economics, Elsevier, vol. 78(C), pages 119-126.
    6. Liu, Ce, 2023. "Stability in repeated matching markets," Theoretical Economics, Econometric Society, vol. 18(4), November.
    7. 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.
    8. Yannai A. Gonczarowski & Ori Heffetz & Clayton Thomas, 2022. "Strategyproofness-Exposing Mechanism Descriptions," Papers 2209.13148, arXiv.org, revised Jul 2023.
    9. Guillaume Haeringer & Vincent Iehlé, 2019. "Two-Sided Matching with (Almost) One-Sided Preferences," American Economic Journal: Microeconomics, American Economic Association, vol. 11(3), pages 155-190, August.
    10. Sperisen, Benjamin & Wiseman, Thomas, 2020. "Too good to fire: Non-assortative matching to play a dynamic game," Games and Economic Behavior, Elsevier, vol. 124(C), pages 491-511.
    11. Ortega, Josué, 2019. "The losses from integration in matching markets can be large," Economics Letters, Elsevier, vol. 174(C), pages 48-51.
    12. Yannai A. Gonczarowski & Clayton Thomas, 2022. "Structural Complexities of Matching Mechanisms," Papers 2212.08709, arXiv.org, revised Mar 2024.
    13. Robert Aue & Thilo Klein & Josue Ortega, 2020. "What Happens when Separate and Unequal School Districts Merge?," Papers 2006.13209, arXiv.org.
    14. Kenny Peng & Nikhil Garg, 2023. "Monoculture in Matching Markets," Papers 2312.09841, arXiv.org.
    15. Josu'e Ortega, 2018. "The Losses from Integration in Matching Markets can be Large," Papers 1810.10287, arXiv.org.
    16. Alvin E. Roth, 2023. "Market Design and Maintenance," NBER Chapters, in: New Directions in Market Design, National Bureau of Economic Research, Inc.
    17. Estelle Cantillon, 2017. "Broadening the market design approach to school choice," Oxford Review of Economic Policy, Oxford University Press and Oxford Review of Economic Policy Limited, vol. 33(4), pages 613-634.
    18. Guillaume Haeringer & Vincent Iehlé, 2019. "Two-Sided Matching with (Almost) One-Sided Preferences," American Economic Journal: Microeconomics, American Economic Association, vol. 11(3), pages 155-190, August.
    19. Gonczarowski, Yannai A. & Nisan, Noam & Ostrovsky, Rafail & Rosenbaum, Will, 2019. "A stable marriage requires communication," Games and Economic Behavior, Elsevier, vol. 118(C), pages 626-647.
    20. Ashlagi, Itai & Nikzad, Afshin & Romm, Assaf, 2019. "Assigning more students to their top choices: A comparison of tie-breaking rules," Games and Economic Behavior, Elsevier, vol. 115(C), pages 167-187.

    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:inm:ormoor:v:44:y:2019:i:1:p:122-146. 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: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    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.