Gale-Shapley Stable Marriage Problem Revisited: Strategic Issues and Applications

This paper is motivated by a study of the mechanism used to assign primary school students in Singapore to secondary schools. The assignment process requires that primary school students submit a rank ordered list of six schools to the Ministry of Education. Students are then assigned to secondary schools based on their preferences, with priority going to those with the highest examination scores on the Primary School Leaving Examination (PSLE). The current matching mechanism is plagued by several problems, and a satisfactory resolution of these problems necessitates the use of a stable matching mechanism. In fact, the student-optimal and school-optimal matching mechanisms of Gale and Shapley [2] are natural candidates. Stable matching problems were first studied by Gale and Shapley [2]. In a stable marriage problem we have two finite sets of players, conveniently called the set of men (M) and the set of women (W ). We assume that every member of each set has strict preferences over the members of the opposite sex. In the rejection model, the preference list of a player is allowed to be incomplete in the sense that players have the option of declaring some of the members of the opposite sex as unacceptable; in the Gale-Shapley model we assume that preference lists of the players are complete. A matching is just a one-to one mapping between the two sexes; in the rejection model, we also include the possibility that a player may be unmatched, i.e. the player’s assigned partner in the matching is himself/herself. The matchings of interest to us are those with the crucial stability property, defined as follows: A matching μ is said to be unstable if there is a man-woman pair, who both prefer each other to their (current) assigned partners in μ; this pair is said to block the matching μ, and is called a blocking pair for μ. A stable matching is a matching that is not unstable. The significance of stability is best highlighted by a system where acceptance of the proposed matching is