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].

中文翻译：

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具有一致偏好和合作伙伴的多方面匹配市场

我们为博士学位市场的学生，顾问和共同顾问介绍了三边稳定匹配问题的一种变体。在我们的形式化过程中，学生对顾问和共同顾问的偏好是（词典顺序），而后者仅对学生具有偏好（因此顾问和共同顾问是合作的）。一个学生必须与一名顾问和一名联合顾问相匹配，或者根本不匹配。与以前的工作相比，顾问-学生和学生-共同顾问对可能互不接受，例如，学生可能不想与顾问或共同顾问合作，反之亦然。我们证明了稳定的三边匹配始终存在，并且提出了PhD算法，这是一种具有多项式运行时间的三边匹配算法，该算法使用任何两边的稳定匹配算法作为匹配引擎。从双面市场的结果中借用，我们提供了一些近似最优结果。我们还介绍了配额扩展的三边市场，其中每个学生进行多个项目，每个项目都由一名顾问和一名联合顾问进行监督。由于在实践中通常情况下，同一学生不应与同一顾问或共同顾问共同完成一个以上的项目，因此我们针对这种情况修改了PhD算法，方法是将两面的Gale-Shapley算法改编为一对多的双边市场，其中同一对最多可以匹配一次。我们还将三边市场概括为由n-1美元的两面市场组成的$ n $面市场。我们将PhD算法扩展到此多面设置，以计算多项式时间内的稳定匹配，然后讨论将其扩展到任意配额的情况。最后，