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Revisiting stability in one-to-one matching problems

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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.
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    3. Kesten, Onur & Unver, Utku, 2015. "A theory of school choice lotteries," Theoretical Economics, Econometric Society, vol. 10(2), May.
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    5. Robert W. Irving, 2008. "Stable matching problems with exchange restrictions," Journal of Combinatorial Optimization, Springer, vol. 16(4), pages 344-360, November.
    6. Ehlers, Lars, 2007. "Von Neumann-Morgenstern stable sets in matching problems," Journal of Economic Theory, Elsevier, vol. 134(1), pages 537-547, May.
    7. 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.
    8. 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.
    9. 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.
    10. 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.
    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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