Coalition Manipulations in the Gale-Shapley Algorithm

It is well-known that the Gale-Shapley algorithm is not truthful for all agents. Previous studies in this category concentrate on manipulations using incomplete preference lists by a single woman and by the set of all women. Little is known about manipulations by a subset of women or other types of manipulations, such as permutation of complete preference lists. In this paper, we consider manipulations by any subset of women with arbitrary preferences (either incomplete or complete). For the setting where agents can report an incomplete preference list (aka. general manipulations), we show that a strong Nash equilibrium of the induced manipulation game always exists among the manipulators and the equilibrium outcome is unique and Pareto-dominant. In addition, the set of matchings achievable by manipulations has a lattice structure. For the setting where agents can only report complete preference lists (aka. permutation manipulations), we give answers to Gusfield and Irving’s open question on what matchings can be achieved in the induced manipulation games. We first construct a counter-example to show that a Pareto-dominant outcome may not exist. Then we present a polynomial-time algorithm to find a Pareto-optimal strategy profile for the induced manipulation game. Furthermore, we show that Pareto-optimality is equivalent to super-strong Nash equilibrium outcomes and all such matchings can be found by our algorithm. The results for the second part are enabled by connecting this problem to the stable roommate problem and using techniques there to analyze a graph called suitor graph. We also introduce several new concepts, such as maximum rotation and principle set, and develop a series of original techniques. Even though all these results may suggest that the Gale-Shapley algorithm is vulnerable to coalition manipulations, we do, however, prove a hardness result in the end, saying that it is NP-complete to find a manipulation that induces a matching strictly better off for all manipulators.