We study situations of allocating positions to students based on priorities. An example is the assignment of medical students to hospital residencies on the basis of 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 in Econometrica 70:2489-2497, 2002). We show that in the presence of couples acyclicity is still necessary, but not sufficient. A second necessary condition 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 and show that it is a sufficient condition for the existence of a fair and efficient placement mechanism. 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.
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. We identify a sufficient condition such that the so-called sequential placement mechanism produces a fair and efficient allocation.