Manipulation Strategies for the Rank-Maximal Matching Problem

Abstract

We consider manipulation strategies for the rank-maximal matching problem. Let G = (A ∪ P, E) be a bipartite graph such that A denotes a set of applicants and P a set of posts. Each applicant a ∈ A has a preference list over the set of his neighbours in G, possibly involving ties. A matching M is any subset of edges from E such that no two edges of M share an endpoint. A rank-maximal matching is one in which the maximum number of applicants is matched to their rank one posts, subject to this condition, the maximum number of applicants is matched to their rank two posts and so on. A central authority matches applicants to posts in G using one of rank-maximal matchings. Let a1 be the sole manipulative applicant, who knows the preference lists of all the other applicants and wants to falsify his preference list, so that, he has a chance of getting better posts than if he were truthful, i.e., than if he gave a true preference list. We give three manipulation strategies for a1 in this paper. In the first problem 'best nonfirst', the manipulative applicant a1 wants to ensure that he is never matched to any post worse than the most preferred post among those of rank greater than one and obtainable, when he is truthful. In the second strategy 'min max' the manipulator wants to construct a preference list for a1 such that the worst post he can become matched to by the central authority is best possible or in other words, a1 wants to minimize the maximal rank of a post he can become matched to. To be able to carry out strategy 'best nonfirst', a1 only needs to know the most preferred post of each applicant, whereas putting into effect 'min max' requires the knowledge of whole preference lists of all applicants. The last manipulation strategy 'improve best' guarantees that a1 is matched to his most preferred post at least in some rank-maximal matchings.