IDEAS home Printed from https://ideas.repec.org/p/nwu/cmsems/1247.html
   My bibliography  Save this paper

Stable Matchings and the Small Core in Nash Equilibrium in the College Admissions Problem

Author

Listed:
  • Jinpeng Ma

Abstract

Both rematching proof and strong equilibrium outcomes are stable with respect to the true preferences in the marriage problem. We show that not all rematching proof or strong equilibrium outcomes are stable in the college admissions problem. But we show that both rematching proof and strong equilibrium outcomes with truncations at the match point are all stable in the college admissions problem. Further, all true stable matchings can be achieved in both rematching proof and strong equilibrium with truncations at the match point. We show that any Nash equilibrium in truncations admits one and only one matching, stable or not. Therefore, the core at a Nash equilibrium in truncations must be small. But examples exist such that the set of stable matchings with respect to a Nash equilibrium may contain more than one matching. Nevertheless, each Nash equilibrium can only admit at most one true stable matching. If, indeed, there is a true stable matching at a Nash equilibrium, then the only possible equilibrium outcome will be the true stable matching, no matter how players manipulate their equilibrium strategies and how many other unstable matchings are there at the Nash equilibrium. Thus, we show that a necessary and sufficient condition for the stable matching rule to be implemented in a subset of Nash equilibria by a direct revelation game induced by a stable matching mechanism is that every Nash equilibrium profile in that subset admits one and only one true stable matching.

Suggested Citation

  • Jinpeng Ma, 1998. "Stable Matchings and the Small Core in Nash Equilibrium in the College Admissions Problem," Discussion Papers 1247, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  • Handle: RePEc:nwu:cmsems:1247
    as

    Download full text from publisher

    File URL: http://www.kellogg.northwestern.edu/research/math/papers/1247.pdf
    File Function: main text
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Danilov, Vladimir, 1992. "Implementation via Nash Equilibria," Econometrica, Econometric Society, vol. 60(1), pages 43-56, January.
    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. 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.

    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. Mezzetti, Claudio & Renou, Ludovic, 2012. "Implementation in mixed Nash equilibrium," Journal of Economic Theory, Elsevier, vol. 147(6), pages 2357-2375.
    2. Murat R. Sertel & M. Remzi Sanver, 2004. "Strong equilibrium outcomes of voting games ¶are the generalized Condorcet winners," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 22(2), pages 331-347, April.
    3. Ahmed Doghmi & Abderrahmane Ziad, 2013. "Nash Implementation in Private Good Economies with Single-Plateaued Preferences," Economics Working Paper Archive (University of Rennes & University of Caen) 201311, Center for Research in Economics and Management (CREM), University of Rennes, University of Caen and CNRS.
    4. Xiong, Siyang, 2021. "Designing referenda: An economist's pessimistic perspective," Journal of Economic Theory, Elsevier, vol. 191(C).
    5. Maskin, Eric & Sjostrom, Tomas, 2002. "Implementation theory," Handbook of Social Choice and Welfare,in: K. J. Arrow & A. K. Sen & K. Suzumura (ed.), Handbook of Social Choice and Welfare, edition 1, volume 1, chapter 5, pages 237-288 Elsevier.
    6. Sonmez, Tayfun, 1996. "Implementation in generalized matching problems," Journal of Mathematical Economics, Elsevier, vol. 26(4), pages 429-439.
    7. Dirk Bergemann & Stephen Morris & Olivier Tercieux, 2012. "Rationalizable Implementation," World Scientific Book Chapters, in: Robust Mechanism Design The Role of Private Information and Higher Order Beliefs, chapter 11, pages 375-404, World Scientific Publishing Co. Pte. Ltd..
    8. Corchón, Luis C., 2008. "The theory of implementation : what did we learn?," UC3M Working papers. Economics we081207, Universidad Carlos III de Madrid. Departamento de Economía.
    9. Ahmed Doghmi & Abderrahmane ZIAD, 2012. "On Partial Honesty Nash Implementation," Economics Working Paper Archive (University of Rennes & University of Caen) 201201, Center for Research in Economics and Management (CREM), University of Rennes, University of Caen and CNRS.
    10. Sang-Chul Suh, 1996. "An algorithm for checking strong Nash implementability," Journal of Mathematical Economics, Elsevier, vol. 25(1), pages 109-122.
    11. Ahmed Doghmi, 2013. "Nash Implementation in an Allocation Problem with Single-Dipped Preferences," Games, MDPI, vol. 4(1), pages 1-12, January.
    12. Klaus Nehring & Massimiliano Marcellino, 2003. "Monotonicity Implies Strategy-Proofness For Correspondences," Working Papers 193, University of California, Davis, Department of Economics.
    13. Olivier Bochet, 2007. "Nash Implementation with Lottery Mechanisms," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 28(1), pages 111-125, January.
    14. Diss, Mostapha & Doghmi, Ahmed & Tlidi, Abdelmonaim, 2016. "Strategy proofness and unanimity in many-to-one matching markets," MPRA Paper 75927, University Library of Munich, Germany, revised 08 Dec 2016.
    15. Bilge Yilmaz & Murat R. Sertel, 1999. "The majoritarian compromise is majoritarian-optimal and subgame-perfect implementable," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 16(4), pages 615-627.
    16. Matthew O. Jackson, 2001. "A crash course in implementation theory," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 18(4), pages 655-708.
    17. Bezalel Peleg & Ariel Procaccia, 2010. "Implementation by mediated equilibrium," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(1), pages 191-207, March.
    18. Doğan, Battal, 2017. "Eliciting the socially optimal allocation from responsible agents," Journal of Mathematical Economics, Elsevier, vol. 73(C), pages 103-110.
    19. Michele Lombardi & Naoki Yoshihara, 2013. "A full characterization of nash implementation with strategy space reduction," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 54(1), pages 131-151, September.
    20. Benoit, Jean-Pierre & Ok, Efe A., 2006. "Maskin's Theorem with limited veto power," Games and Economic Behavior, Elsevier, vol. 55(2), pages 331-339, May.

    More about this item

    JEL classification:

    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory
    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations

    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:nwu:cmsems:1247. 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: Fran Walker (email available below). General contact details of provider: https://edirc.repec.org/data/cmnwuus.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.