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Finding Stable Matchings in PhD Markets with Consistent Preferences and Cooperative Partners

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  • Maximilian Mordig
  • Riccardo Della Vecchia
  • Nicol`o Cesa-Bianchi
  • Bernhard Scholkopf

Abstract

We introduce a new algorithm for finding stable matchings in multi-sided matching markets. Our setting is motivated by a PhD market of students, advisors, and co-advisors, and can be generalized to supply chain networks viewed as $n$-sided markets. In the three-sided PhD market, students primarily care about advisors and then about co-advisors (consistent preferences), while advisors and co-advisors have preferences over students only (hence they are cooperative). A student must be matched to one advisor and one co-advisor, or not at all. In contrast to previous work, advisor-student and student-co-advisor pairs may not be mutually acceptable (e.g., a student may not want to work with an advisor or co-advisor and vice versa). We show that three-sided stable matchings always exist, and present an algorithm that, in time quadratic in the market size (up to log factors), finds a three-sided stable matching using any two-sided stable matching algorithm as matching engine. We illustrate the challenges that arise when not all advisor-co-advisor pairs are compatible. We then generalize our algorithm to $n$-sided markets with quotas and show how they can model supply chain networks. Finally, we show how our algorithm outperforms the baseline given by [Danilov, 2003] in terms of both producing a stable matching and a larger number of matches on a synthetic dataset.

Suggested Citation

  • Maximilian Mordig & Riccardo Della Vecchia & Nicol`o Cesa-Bianchi & Bernhard Scholkopf, 2021. "Finding Stable Matchings in PhD Markets with Consistent Preferences and Cooperative Partners," Papers 2102.11834, arXiv.org, revised Jul 2021.
  • Handle: RePEc:arx:papers:2102.11834
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    References listed on IDEAS

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    1. Danilov, V. I., 2003. "Existence of stable matchings in some three-sided systems," Mathematical Social Sciences, Elsevier, vol. 46(2), pages 145-148, October.
    2. Roth, Alvin E., 1989. "Two-sided matching with incomplete information about others' preferences," Games and Economic Behavior, Elsevier, vol. 1(2), pages 191-209, June.
    3. , & ,, 2006. "A theory of stability in many-to-many matching markets," Theoretical Economics, Econometric Society, vol. 1(2), pages 233-273, June.
    4. Roth, Alvin E, 1984. "The Evolution of the Labor Market for Medical Interns and Residents: A Case Study in Game Theory," Journal of Political Economy, University of Chicago Press, vol. 92(6), pages 991-1016, December.
    5. Alvin E. Roth, 1985. "Conflict and Coincidence of Interest in Job Matching: Some New Results and Open Questions," Mathematics of Operations Research, INFORMS, vol. 10(3), pages 379-389, August.
    6. Maximilian Mordig & Riccardo Della Vecchia & Nicol`o Cesa-Bianchi & Bernhard Scholkopf, 2021. "Two-Sided Matching Markets in the ELLIS 2020 PhD Program," Papers 2101.12080, arXiv.org, revised Mar 2021.
    7. Roth, Alvin E, 1991. "A Natural Experiment in the Organization of Entry-Level Labor Markets: Regional Markets for New Physicians and Surgeons in the United Kingdom," American Economic Review, American Economic Association, vol. 81(3), pages 415-440, June.
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