On Likely Solutions of a Stable Marriage Problem

Abstract

An ($n$ men-$n$ women) stable marriage problem is studied under the assumption that the individual preferences for a marriage partner are uniformly random and mutually independent. We show that the total number of stable matchings (marriages) is at least $(n/\log n)^{1/2}$ with high probability (whp) as $n \rightarrow \infty$ and also that the total number of stable marriage partners of each woman (man) is asymptotically normal with mean and variance close to $\log n$. It is proved that in the male (female) optimal stable marriage the largest rank of a wife (husband) is whp of order $\log^2 n$, while the largest rank of a husband (wife) is asymptotic to $n$. Earlier, we proved that for either of these extreme matchings the total rank is whp close to $n^2/\log n$. Now, we are able to establish a whp existence of an egalitarian marriage for which the total rank is close to $2n^{3/2}$ and the largest rank of a partner is of order $n^{1/2} \log n$. Quite unexpectedly, the stable matchings obey, statistically, a "law of hyperbola": namely, whp the product of the sum of husbands' ranks and the sum of wives' ranks in a stable matching turns out to be asymptotic to $n^3$, uniformly over all stable marriages. The key elements of the proofs are extensions of the McVitie-Wilson proposal algorithm and of Knuth's integral formula for the probability that a given matching is stable, and also a notion of rotations due to Irving. Methods developed in this paper may, in our opinion, be found useful in probabilistic analysis of other combinatorial algorithms.