We introduce a variant of the three-sided stable matching problem for a PhD market with students, advisors, and co-advisors . In our formalization, students have consistent (lexicographic) preferences over advisors and co-advisors , and the latter have preferences over students only (hence advisors and co-advisors 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 stable three-sided  matchings always exist, and present the PhD algorithm , a three-sided matching algorithm with polynomial running time which uses any two-sided stable matching algorithm as matching engine. Borrowing from results on two-sided markets, we provide some approximate optimality results. We also present an extension to three-sided markets with quotas , where each student conducts several projects, and each project is supervised by one advisor and one co-advisor. As it is often the case in practice that the same student should not do more than one project with the same advisor or co-advisor, we modify our PhD algorithm for this setting by adapting the two-sided Gale--Shapley algorithm to many-to-many two-sided markets, in which the same pair can match at most once. We also generalize the three-sided market to an n-sided market consisting of n-1 two-sided markets. We extend the PhD algorithm to this multi-sided setting to compute ]  a stable matching in polynomial time, and we discuss its extension to arbitrary quotas. Finally, we illustrate the challenges that arise when not all advisor-co-advisor pairs are compatible, and critically review the statements from 30, 29 .
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 .
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