Successful Manipulation in Stable Marriage Model with Complete Preference Lists

Abstract

SUMMARYThis paper deals with a strategic issue in the stable marriage model with complete preference lists (i.e., a preference list of an agent is a permutation of all the members of the opposite sex). Given complete preference lists of n men over n women, and a marriage µ, we consider the problem for finding preference lists of n women over n men such that the men-proposing deferred acceptance algorithm (Gale-Shapley algorithm) adopted to the lists produces µ. We show a simple necessary and sufficient condition for the existence of a set of preference lists of women over men. Our condition directly gives an O(n 2 ) time algorithm for finding a set of preference lists, if it exists.

Preference profiles determining the proposals in the Gale–Shapley algorithm for stable matching problems

“…Now let L * M be an arbitrary preference profile of men satisfying (9), and consider the existence problem of Kobayashi and Matsui [7] or the first problem in [8]. If their polynomial time algorithm provides a preference profile of women L * W that together with L * M makes the mGS produce f , we are done.…”

“…Matsui's work has been published in [7], where given a preference profile of men and a complete matching, they consider the problem of finding a preference profile of women such that the men-proposing GaleShapley algorithm produces the given complete matching, and show that the problem is solvable in polynomial time. They also consider several variations of the problem in [8], and show that one of them results in NP-completeness.…”

“…When we are given a preference profile of men, being given a complete matching is equivalent to being given the set of women to whom each man proposes to during the execution of the algorithm. Hence the problem that Kobayashi and Matsui consider in [7], which is the first problem in [8], can be restated as "given a preference profile of men and a set of proposals, find a preference profile of women such that during the execution of the men-proposing Gale-Shapley algorithm each man proposes to women prescribed by the set of proposals." The aim of this paper is to answer the question what if we switch the roles of men and women in this setting.…”

Finding Witnesses for Stability in the Hospitals/Residents Problem

“…We implement these algorithms and compare their computation time and accuracy using random instances. Related work: For SM, Kobayashi and Matsui [4] have previously presented a similar issue and studied it in the context of strategic issue: Given men's complete preference lists and a matching, they consider the problem of finding women's complete preference lists that make a given matching man-optimal stable. Organization of the paper: This paper is organized as follows.…”

Cheating Strategies for the Gale-Shapley Algorithm with Complete Preference Lists

“…In [6], the authors discussed the following problem; given complete preference lists of men and a marriage μ, find a set of complete preference lists of women such that the men-proposing Gale-Shapley algorithm applied to the lists produces μ. We established a simple necessary and sufficient condition for the existence of such a set of complete preference lists.…”