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Revisiting stability in one-to-one matching problems

Author

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  • Wouter Vergote

    (Columbia University
    CEREC, Université Saint-Louis-Brussels)

Abstract

This paper studies the stability of a status quo matching by considering the set of matching locations as a primitive of a one-to-one matching problem, alongside the agents and their preferences. As such we generalize the approach of Morrill (J Econ Theory 145:1739–1756, 2010) who was the first to study matching problems with location restrictions. We develop two novel stability concepts, direct and (coalition-) trade stability, akin to Gale–Shapley stability and Alcalde’s (Econ Des 1:275–287, 1995) concept of exchange stability, respectively, and derive connections with existing stability concepts. We show that coalition-trade stability is a refinement of direct stability. We then demonstrate that when there are no matching restrictions, direct stability is equivalent to Gale–Shapley stability and coalition-trade stability is equivalent to requiring both exchange stability and Gale–Shapley stability. In addition, we reveal a link between trade dominance and indirect dominance, Harsanyi’s farsighted dominance concept. For the class of individually rational matching problems, we show that indirect dominance is a refinement of trade dominance. However, these two dominance notions do not always generate the same stable (set of) matchings.

Suggested Citation

  • Wouter Vergote, 2019. "Revisiting stability in one-to-one matching problems," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 7(1), pages 59-75, May.
    • Handle: RePEc:spr:etbull:v:7:y:2019:i:1:d:10.1007_s40505-018-0143-x
    DOI: 10.1007/s40505-018-0143-x
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    References listed on IDEAS

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    1. , & , J. & ,, 2011. "Von Neumann-Morgenstern farsightedly stable sets in two-sided matching," Theoretical Economics, Econometric Society, vol. 6(3), September.
    2. Kesten, Onur & Unver, Utku, 2015. "A theory of school choice lotteries," Theoretical Economics, Econometric Society, vol. 10(2), May.
    3. Robert W. Irving, 2008. "Stable matching problems with exchange restrictions," Journal of Combinatorial Optimization, Springer, vol. 16(4), pages 344-360, November.
    4. Ehlers, Lars, 2007. "Von Neumann-Morgenstern stable sets in matching problems," Journal of Economic Theory, Elsevier, vol. 134(1), pages 537-547, May.
    5. Bettina Klaus & Flip Klijn & Markus Walzl, 2011. "Farsighted Stability for Roommate Markets," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 13(6), pages 921-933, December.
    6. José Alcalde, 1994. "Exchange-proofness or divorce-proofness? Stability in one-sided matching markets," Review of Economic Design, Springer;Society for Economic Design, vol. 1(1), pages 275-287, December.
    7. John C. Harsanyi, 1974. "An Equilibrium-Point Interpretation of Stable Sets and a Proposed Alternative Definition," Management Science, INFORMS, vol. 20(11), pages 1472-1495, July.
    8. Effrosyni Diamantoudi & Licun Xue, 2003. "Farsighted stability in hedonic games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 21(1), pages 39-61, August.
    9. Debraj Ray & Rajiv Vohra, 2015. "The Farsighted Stable Set," Econometrica, Econometric Society, vol. 83(3), pages 977-1011, May.
    10. Iñarra, E. & Larrea, C. & Molis, E., 2013. "Absorbing sets in roommate problems," Games and Economic Behavior, Elsevier, vol. 81(C), pages 165-178.
    11. Morrill, Thayer, 2010. "The roommates problem revisited," Journal of Economic Theory, Elsevier, vol. 145(5), pages 1739-1756, September.
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    Cited by:

    1. Duygu Nizamogullari & İpek Özkal-Sanver, 2022. "A note on roommate problems with a limited number of rooms," Review of Economic Design, Springer;Society for Economic Design, vol. 26(4), pages 553-560, December.

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    More about this item

    Keywords

    One-to-one matching; Direct dominance; Trade dominance; Indirect dominance;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory

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