Near Feasible Stable Matchings with Complementarities

Abstract

The National Resident Matching program strives for a stable matching of medical students to teaching hospitals. With the presence of couples, stable matchings need not exist. For any student preferences, we show that each instance of a stable matching problem has a ’nearby’ instance with a table matching. The nearby instance is obtained by perturbing the capacities of the hospitals. Specifically, given a reported capacity k_h ( for each hospital h, we find a redistribution of the slot capacities k¹_h satisfying [k_h–k¹_h] ≤ 4 for all hospital h, and ∑_h k_h ≤ ∑ k¹_h ≤ ∑_h k_h 9, such that a stable matching exists with respect to k¹. Our approach is general and applies to other type of complementarities, as well as matchings with side constraints and contracts.

Keywords: stable matching, complementarities, Scarf's lemma

JEL Classification: C78, D47