Paths to marriage stability

Abstract

We obtain a family of algorithms that determine stable matchings for the stable marriage problem by starting with an arbitrary matching and iteratively satisfying blocking pairs, that is, matching couples who both prefer to be together over the outcome of the current matching. The existence of such an algorithm is related to a question raised by Knuth (1976) and was recently resolved positively by Roth and Vande Vate (1992). The basic version of our method depends on a fixed ordering of all mutually acceptable man-woman pairs which is consistent with the preferences of either all men or of all women. Given such an ordering, we show that starting with an arbitrary matching and iteratively satisfying the highest blocking pair at each iteration will eventually yield a stable matching. We show that the single-proposal variant of the Gale-Shapley algorithm as well as the Roth-Vande Vate algorithm are instances of our approach. We also demonstrate that an arbitrary decentralized system does not guarantee convergence to a stable matching.