Matchings (Stable Marriages)

Abstract

Given a set of men and a set of women, a matching is a set of pairs, each pair containing one man and one woman, such that no person is in more than one pair. We shall be interested in finding matchings satisfying various criteria. The first problem we’ll consider is called the stable marriage problem. We assume that there are the same number of men as women, and that each person ranks the people of the opposite sex in order of preference. A matching is stable if there is no unmatched pair {a,b} such that both a and b prefer each other to their present partners. (If such a pair existed, they would run off together.) Though we speak of men and women, this is actually a rather facetious viewpoint; this problem is typically applied to relationships somewhat more pragmatic than marriage, such as roommate assignments, dormitory room assignments, and university admissions. Nevertheless, here we’ll discuss the problem in terms of marriages between men and women.