Stable Nash Equilibria in the Gale-Shapley Matching Game

In this article we study the stable marriage game induced by the men-proposing GaleShapley algorithm. Our setting is standard: all the lists are complete and the matching mechanism is the men-proposing Gale-Shapley algorithm. It is well known that in this setting, men cannot cheat, but women can. In fact, Teo, Sethuraman and Tan [TST01], show that there is a polynomial time algorithm to obtain, for a given strategy (the set of all lists) Q and a woman w, the best partner attainable by changing her list. However, what if the resulting matching is not stable with respect to Q? Obviously, such a matching would be vulnerable to further manipulation, but is not mentioned in [TST01]. In this paper, we consider (safe) manipulation that implies a stable matching in a most general setting. Specifically, our goal is to decide for a given Q, if w can manipulate her list to obtain a strictly better partner with respect to the true strategy P (which may be different from Q), and also the outcome is a stable matching for P .