The Uniqueness of Stable Matchings

Abstract

This paper analyses conditions on agents' preferences for a unique stable matching in models of two-sided matching with non-transferable utility. The No Crossing Condition (NCC) is sufficient for uniqueness; it is based on the notion that a person's characteristics, for example their personal qualities or their productive capabilities, not only form the basis of their own attraction to the opposite sex but also determine their own preferences. The paper also shows that a weaker condition, alpha-reducibility, is both necessary and sufficient for a population and any of its subpopulations to have a unique stable matching. If preferences are based on utility functions with agents' characteristics as arguments, then the NCC may be easy to verify. The paper explores conditions on utility functions which imply that the NCC is satisfied whatever the distribution of characteristics. The usefulness of this approach is illustrated by two simple models of household formation.