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On random exchange-stable matchings

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  • Pittel, Boris

Abstract

Consider the group of n men and n women, each with their own preference list for a potential marriage partner. The stable marriage is a bipartite matching such that no unmatched pair (man, woman) prefer each other to their partners in the matching. Its non-bipartite version, with an even number n of members, is known as the stable roommates problem. Jose Alcalde introduced an alternative notion of exchange-stable, one-sided, matching: no two members prefer each other’s partners to their own partners in the matching. Katarina Cechlárová and David Manlove showed that the e-stable matching decision problem is NP-complete for both types of matchings. We prove that the expected number of e-stable matchings is asymptotic to πn21∕2 for two-sided case, and to e1∕2 for one-sided case. However, the standard deviation of this number exceeds 1.13n, (1.06n resp.). As an obvious byproduct, there exist instances of preference lists with at least 1.13n (1.06n resp.) e-stable matchings. The probability that there is no matching which is stable and e-stable is at least 1−e−n1∕2−o(1), (1−O(2−n∕2) resp.).

Suggested Citation

  • Pittel, Boris, 2018. "On random exchange-stable matchings," Mathematical Social Sciences, Elsevier, vol. 93(C), pages 1-13.
    • Handle: RePEc:eee:matsoc:v:93:y:2018:i:c:p:1-13
    DOI: 10.1016/j.mathsocsci.2018.01.002
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    References listed on IDEAS

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    1. Robert W. Irving, 2008. "Stable matching problems with exchange restrictions," Journal of Combinatorial Optimization, Springer, vol. 16(4), pages 344-360, November.
    2. 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.
    3. 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.
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