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On the Convergence of Swap Dynamics to Pareto-Optimal Matchings

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Web and Internet Economics (WINE 2019)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 11920))

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Abstract

We study whether Pareto-optimal stable matchings can be reached via pairwise swaps in one-to-one matching markets with initial assignments. We consider housing markets, marriage markets, and roommate markets as well as three different notions of swap rationality. Our main results are as follows. While it can be efficiently determined whether a Pareto-optimal stable matching can be reached when defining swaps via blocking pairs, checking whether this is the case for all such sequences is computationally intractable. When defining swaps such that all involved agents need to be better off, even deciding whether a Pareto-optimal stable matching can be reached via some sequence is intractable. This confirms and extends a conjecture made by Damamme et al. (2015), who have furthermore shown that convergence to a Pareto-optimal matching is guaranteed in housing markets with single-peaked preferences. We show that in marriage and roommate markets, single-peakedness is not sufficient for this to hold, but the stronger restriction of one-dimensional Euclidean preferences is.

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Notes

  1. 1.

    Once the old partners are alone, they have an incentive to form a new pair. Roth and Vande Vate [26] therefore decompose \( BP \)-swaps into two steps. We do not explicitly consider these steps in order to always maintain a perfect matching [cf. 23].

  2. 2.

    Assuming that the old partners also form a new pair does not alter this result.

  3. 3.

    The presence of a global ranking over all possible pairs does not imply that preferences are 1-Euclidean. Consider for instance, in roommate markets, the following preference profile: \(1: 2\succ 3\succ 4\), \(2: 1\succ 4\succ 3\), \(3: 4\succ 1\succ 2\), \(4: 3\succ 2\succ 1\).

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Authors and Affiliations

  1. Technical University of Munich, 80538, Munich, Germany

    Felix Brandt & Anaëlle Wilczynski

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  1. Felix Brandt
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  2. Anaëlle Wilczynski
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Correspondence to Felix Brandt .

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Editors and Affiliations

  1. University of Patras, Rio, Greece

    Ioannis Caragiannis

  2. Google Research New York, New York, NY, USA

    Vahab Mirrokni

  3. The University of Texas System, Austin, TX, USA

    Evdokia Nikolova

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Brandt, F., Wilczynski, A. (2019). On the Convergence of Swap Dynamics to Pareto-Optimal Matchings. In: Caragiannis, I., Mirrokni, V., Nikolova, E. (eds) Web and Internet Economics. WINE 2019. Lecture Notes in Computer Science(), vol 11920. Springer, Cham. https://doi.org/10.1007/978-3-030-35389-6_8

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  • DOI: https://doi.org/10.1007/978-3-030-35389-6_8

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