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

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