Paths to marriage stability

AbstractWe 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

Improved Paths to Stability for the Stable Marriage Problem

The stable marriage problem requires one to find a marriage with no blocking pair. Given a matching that is not stable, Roth and Vande Vate have shown that there exists a sequence of matchings that leads to a stable matching in which each successive matching is obtained by satisfying a blocking pair. The sequence produced by Roth and Vande Vate's algorithm is of length $O(n^3)$ where $n$ is the number of men (and women). In this paper, we present an algorithm that achieves stability in a sequence of matchings of length $O(n^2)$. We also give an efficient algorithm to find the stable matching closest to the given initial matching under an appropriate distance function between matchings

Preference Structure and Random Paths to Stability in Matching Markets

This paper examines how preference correlation and intercorrelation combine to influence the length of a decentralized matching market's path to stability. In simulated experiments, marriage markets with various preference specifications begin at an arbitrary matching of couples and proceed toward stability via the random mechanism proposed by Roth and Vande Vate (1990). The results of these experiments reveal that fundamental preference characteristics are critical in predicting how long the market will take to reach a stable matching. In particular, intercorrelation and correlation are shown to have an exponential impact on the number of blocking pairs that must be randomly satisfied before stability is attained. The magnitude of the impact is dramatically different, however, depending on whether preferences are positively or negatively intercorrelated.

Stochastic Stability for Roommate Markets

We show that for any roommate market the set of stochastically stable matchings coincideswith the set of absorbing matchings. This implies that whenever the core is non-empty (e.g.,for marriage markets), a matching is in the core if and only if it is stochastically stable, i.e., stochastic stability is a characteristic of the core. Several solution concepts have beenproposed to extend the core to all roommate markets (including those with an empty core).An important implication of our results is that the set of absorbing matchings is the onlysolution concept that is core consistent and shares the stochastic stability characteristic withthe core.Economics (Jel: A)