Fair and Efficient Student Placement with Couples

We study situations of allocating positions or jobs to students or workers based on priorities. An example is the assignment of medical students to hospital residencies on the basis of one or several entrance exams. For markets without couples, e.g., for ``undergraduate student placement,'' acyclicity is a necessary and sufficient condition for the existence of a fair and efficient placement mechanism (Ergin, 2002). We show that in the presence of couples, which introduces complementarities into the students' preferences, acyclicity is still necessary, but not sufficient (Theorem 4.1). A second necessary condition (Theorem 4.2) is ``priority-togetherness'' of couples. A priority structure that satisfies both necessary conditions is called pt-acyclic. For student placement problems where all quotas are equal to one we characterize pt-acyclicity (Lemma 5.1) and show that it is a sufficient condition for the existence of a fair and efficient placement mechanism (Theorem 5.1). If in addition to pt- acyclicity we require ``reallocation-'' and ``vacancy-fairness'' for couples, the so-called dictator-bidictator placement mechanism is the unique fair and efficient placement mechanism (Theorem 5.2). Finally, for general student placement problems, we show that pt-acyclicity may not be sufficient for the existence of a fair and efficient placement mechanism (Examples 5.4, 5.5, and 5.6). We identify a sufficient condition such that the so-called sequential placement mechanism produces a fair and efficient allocation (Theorem 5.3).