Coalition Manipulation of Gale-Shapley Algorithm

Weiran Shen, Pingzhong Tang, Yuan Deng

Abstract:

It is well-known that the Gale-Shapley algorithm is not truthful for all agents. Previous studies on this front mostly focus on blacklist manipulations by a single woman and by the set of all women. Little is known about manipulations by a coalition of women or other types of manipulations, such as manipulation by permuting preference lists.
In this paper, we consider the problem of finding an equilibrium for a coalition of women (aka. liars) in the Gale-Shapley algorithm. We restrict attentions on manipulations that induce stable matchings. For the incomplete preference list setting, where liars can truncate their preference lists, we show that a strong Nash equilibrium always exists and the matching from such equilibria is unique. The equilibrium outcome is strongly Pareto dominant for all liars among the set of matchings achievable by manipulation: every woman is matched with the same man as the one she matches in her best single-agent manipulation. For the complete preference list setting where liars can permute their preference list, we first show that a coalition of women can get worse off by manipulating jointly than each performing a single-agent manipulation, thus a strongly Pareto-dominant outcome may not exist by manipulation. We then put forward an efficient algorithm to compute a strong Nash equilibrium that is strongly Pareto-optimal for all liars. We derive connections between the stable marriage problem and stable roommate problem, and use tools there to prove our results for this part. This approach is highly nontrivial and of independent interest.