Marriage Matching: A Conjecture of Donald Knuth

Variations of the Gale-Shapley algorithm have been used and studied extensively in real world markets. Examples include matching medical residents with residency programs, the kidney exchange program and matching college students with on-campus housing. The performance of the Gale-Shapley marriage matching algorithm (1962) has been studied extensively in the special case of men's and women's preferences random. We drop the assumption that women's preferences are random and show that En /n ln n -> 1, where En is the expected number of proposals made when the men-propose Gale-Shapley algorithm is used to match n men with n women. This establishes in spirit a conjecture of Donald Knuth (1976, 1997) of thirty years standing. Under the same assumptions, we also establish bounds on the expected ranking by a woman of her assigned mate. Bounds on men's rankings of their assigned mates follow directly from the conjecture.