The necessary and sufficient condition for the worst-case male optimal stable matching

Abstract

Gale and Shapley [l] introduced and solved the stable matching problem. That problem involves two disjoint sets of equal cardinality n, the men and the women. Each person ranks all members of the opposite sex in order of preference. A stable matching is defined as a complete matching between men and women with the property that no man and woman who are not partners both prefer each other to their actual partners under the matching. Several stable matching algorithms [l-3, S-101 were proposed to solve the problem by returning the male optimal stable solution as its answer. Any sequential one of those algorithms will be called a stable matching algorithm in this paper. The purpose of this paper is to consider the worst-case choice for the sequential stable matching problem. This problem has been investigated by some researchers since the early paper of Wilson [ll]. Itoga [5] presented some conclusions about the nature of the worst-case situation. Tseng and Lee [lo] gave a necessary condition for the worst-case execution of the stable matching problem which takes the maximum number of proposals for the McVitie-Wilson’s algorithm [8,9]. Kapur and Krishnamoorthy [6] presented a worst-case choice which takes the maximum number of stages for Gale-Shapley’s algorithm [l]. In this paper we give the necessary and sufficient condition for the worst-case execution, which leads the sequential stable matching algorithm to take the maximum number of proposals. We then point out that the probability that the worst-case execution occurs when a sequential stable matching algorithm is employed is extremely small.