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Anarchy, stability, and utopia: creating better matchings

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Historically, the analysis of matching has centered on designing algorithms to produce stable matchings as well as on analyzing the incentive compatibility of matching mechanisms. Less attention has been paid to questions related to the social welfare of stable matchings in cardinal utility models. We examine the loss in social welfare that arises from requiring matchings to be stable, the natural equilibrium concept under individual rationality. While this loss can be arbitrarily bad under general preferences, when there is some structure to the underlying graph corresponding to natural conditions on preferences, we prove worst case bounds on the price of anarchy. Surprisingly, under simple distributions of utilities, the average case loss turns out to be significantly smaller than the worst-case analysis would suggest. Furthermore, we derive conditions for the existence of approximately stable matchings that are also close to socially optimal, demonstrating that adding small switching costs can make socially (near-)optimal matchings stable. Our analysis leads to several concomitant results of interest on the convergence of decentralized partner-switching algorithms, and on the impact of heterogeneity of tastes on social welfare.

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Correspondence to Sanmay Das.

Additional information

A preliminary version of this article appeared in SAGT 2009. This work was supported in part by NSF grants CCF-0914782, CNS-1017932, and IIS-0952918. Naamad is now at Princeton University.

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Anshelevich, E., Das, S. & Naamad, Y. Anarchy, stability, and utopia: creating better matchings. Auton Agent Multi-Agent Syst 26, 120–140 (2013).

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