Chennai Mathematical Institute

Seminars




11:45 a.m.
Popular Matchings - structure and cheating strategies

Meghana Nasre
Univ of Texas.
11-03-13


Abstract

We consider the problem of matching a set of agents A to a set of posts P where agents rank posts according to their preferences, possibly involving ties. This problem occurs in several real world scenarios like assigning jobs to applicants, houses to trainees and DVDs to customers. Several notions of optimality have been considered in the literature and we focus on the notion of popularity. 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'.

In this talk, we 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, to improve the set of posts that she gets matched to as opposed to what she got when she was truthful. We show that the optimal cheating strategy for a single agent to get better always can be computed in O(\sqrt{n}m) time when preference lists are allowed to contain ties and in O(m+n) time when preference lists are all strict. Here n = |A| + |P| and m denotes the combined size of all the preference lists.

We also characterize the equilibrium of a non-cooperative game where all agents are manipulative.

To compute the cheating strategies, we develop a switching graph characterization of the popular matchings problem involving ties. The switching graph characterization for the case of ties is of independent interest and answers a part of the open questions posed by McDermid and Irving (J. Comb. Optim. 2011).

These results have been accepted for publication at STACS 2013.





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