Popular Matchings: Structure and Strategic Issues

We consider the strategic issues of the popular matchings problem. Let G = (A ∪ P, E) be a bipartite graph where A denotes a set of agents, P denotes a set of posts and the edges in E are ranked. Each agent ranks a subset of posts in an order of preference, possibly involving ties. A matching M is popular if there exists no matching M ′ such that the number of agents that prefer M ′ to M exceeds the number of agents that prefer M to M . Consider a centralized market where agents submit their preferences and a central authority matches agents to posts according to the notion of popularity. Since a popular matching need not be unique, we assume that the central authority chooses an arbitrary popular matching. Let a1 be the sole manipulative agent who is aware of the true preference lists of all other agents. The goal of a1 is to falsify her preference list to get better always, that is, in the falsified instance (i) every popular matching matches a1 to a post that is at least as good as the most-preferred post that she gets when she was truthful, and (ii) some popular matching matches a1 to a post better than the most-preferred post p that she gets when she was truthful, assuming that p is not one of a1’s (true) most-preferred posts. We show that the optimal cheating strategy for a manipulative agent to get better always can be computed in O(m+n) time when preference lists are all strict and in O( √ nm) time when preference lists are allowed to contain ties. Here n = |A|+ |P| and m = |E|. To compute the cheating strategies, we develop a switching graph characterization of the popular matchings problem involving ties. The switching graph characterization was studied for the case of strict lists by McDermid and Irving (J. Comb. Optim. 2011) and was open for the case of ties. We show an O( √ nm) time algorithm to compute the set of popular pairs using the switching graph. These results are of independent interest and answer a part of the open questions posed by McDermid and Irving.