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original papers : Stable matchings and the small core in Nash equilibrium in the college admissions problem

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  • Jinpeng Ma

    ()
    (Department of Economics, Rutgers University, Camden, NJ 08102, USA)

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    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 in 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 in 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 different are players' equilibrium strategies from the true preferences and how many other unstable matchings are there at that 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 the direct revelation game induced by a stable mechanism is that every Nash equilibrium profile in that subset admits one and only one true stable matching.

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    Bibliographic Info

    Article provided by Springer in its journal Review of Economic Design.

    Volume (Year): 7 (2002)
    Issue (Month): 2 ()
    Pages: 117-134

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    Handle: RePEc:spr:reecde:v:7:y:2002:i:2:p:117-134

    Note: Received: 30 December 1998 / Accepted: 12 October 2001
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    Web page: http://link.springer.de/link/service/journals/10058/index.htm

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    Related research

    Keywords: Stable matchings; Nash equilibrium; college admissions problem;

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    Cited by:
    1. Marilda Sotomayor, 2012. "A further note on the college admission game," International Journal of Game Theory, Springer, vol. 41(1), pages 179-193, February.
    2. Ma, Jinpeng, 2010. "The singleton core in the college admissions problem and its application to the National Resident Matching Program (NRMP)," Games and Economic Behavior, Elsevier, vol. 69(1), pages 150-164, May.

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