Cheating to Get Better Roommates in a Random Stable Matching

This paper addresses strategies for the stable roommates problem, assuming that a stable matching is chosen at random. We investigate how a cheating man should permute his preference list so that he has a higher-ranking roommate probabilistically. In the first part of the paper, we identify a necessary condition for creating a new stable roommate for the cheating man. This condition precludes any possibility of his getting a new roommate ranking higher than all his stable roommates when everyone is truthful. Generalizing to the case that multiple men collude, we derive another impossibility result: given any stable matching in which a subset of men get their best possible roommates, they cannot cheat to create a new stable matching in which they all get strictly better roommates than in the given matching. Our impossibility result, considered in the context of the stable marriage problem, easily re-establishes the celebrated Dubins-Freedman Theorem. The more generalized Demange-Gale-Sotomayor Theorem states that a coalition of men and women cannot cheat to create a stable matching in which everyone of them gets a strictly better partner than in the Gale-Shapley algorithm (with men proposing). We give a sharper result: a coalition of men and women cannot cheat together so that, in a newly-created stable matching, every man in the coalition gets a strictly better partner than in the Gale-Shapley algorithm while none of the women in the coalition is worse off. In the second part of the paper, we present two cheating strategies that guarantee that the cheating man\u27s new probability distribution over stable roommates majorizes the original one. These two strategies do not require the knowledge of the probability distribution of the cheating man. This is important because the problem of counting stable matchings is \#P-complete. Our strategies only require knowing the set of stable roommates that the cheating man has and can be formulated in polynomial time. Our second cheating strategy has an interesting corollary in the context of stable marriage with the Gale-Shapley algorithm. Any woman-optimal strategy will ensure that every woman, cheating or otherwise, ends up with a partner at least as good as when everyone is truthful

Manipulation Strategies for the Rank Maximal Matching Problem

We consider manipulation strategies for the rank-maximal matching problem. In the rank-maximal matching problem we are given a bipartite graph $G = (A \cup P, E)$ such that $A$ denotes a set of applicants and $P$ a set of posts. Each applicant $a \in A$ has a preference list over the set of his neighbours in $G$, possibly involving ties. Preference lists are represented by ranks on the edges - an edge $(a,p)$ has rank $i$, denoted as $rank(a,p)=i$, if post $p$ belongs to one of $a$'s $i$-th choices. A rank-maximal matching is one in which the maximum number of applicants is matched to their rank one posts and subject to this condition, the maximum number of applicants is matched to their rank two posts, and so on. A rank-maximal matching can be computed in $O(\min(c \sqrt{n},n) m)$ time, where $n$ denotes the number of applicants, $m$ the number of edges and $c$ the maximum rank of an edge in an optimal solution. A central authority matches applicants to posts. It does so using one of the rank-maximal matchings. Since there may be more than one rank- maximal matching of $G$, we assume that the central authority chooses any one of them randomly. Let $a_1$ be a 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. In the first problem addressed in this paper the manipulative applicant $a_1$ wants to ensure that he is never matched to any post worse than the most preferred among those of rank greater than one and obtainable when he is truthful. In the second problem the manipulator wants to construct such a preference list that the worst post he can become matched to by the central authority is best possible or in other words, $a_1$ wants to minimize the maximal rank of a post he can become matched to

Cheating to Get Better Roommates in a Random Stable Matching

This paper addresses strategies for the stable roommates problem, assuming that a stable matching is chosen at random. We investigate how a cheating man should permute his preference list so that he has a higher-ranking roommate probabilistically. In the first part of the paper, we identify a necessary condition for creating a new stable roommate for the cheating man. This condition precludes any possibility of his getting a new roommate ranking higher than all his stable roommates when everyone is truthful. Generalizing to the case that multiple men collude, we derive another impossibility result: given any stable matching in which a subset of men get their best possible roommates, they cannot cheat to create a new stable matching in which they all get strictly better roommates than in the given matching. Our impossibility result, considered in the context of the stable marriage problem, easily re-establishes the celebrated Dubins-Freedman Theorem. The more generalized Demange-Gale-Sotomayor Theorem states that a coalition of men and women cannot cheat to create a stable matching in which everyone of them gets a strictly better partner than in the Gale-Shapley algorithm (with men proposing). We give a sharper result: a coalition of men and women cannot cheat together so that, in a newly-created stable matching, every man in the coalition gets a strictly better partner than in the Gale-Shapley algorithm while none of the women in the coalition is worse off