Strategic play in stable marriage problem

The stable marriage problem, as addressed by Gale and Shapely [1] consists of providing a bipartite matching between n “boys” and n “girls” each of whom have a totally ordered preference list over the other set such that there exists no “boy” and no “girl” that would prefer each other over their partner in the matching. In this paper, we analyze the cases of strategic play by the “boys” in the game directly inspired by this problem. We provide an O(n) algorithm for determining a matching which is not necessarily stable in the Gale-Shapely sense, but it is coalition-stable, in that no player has a selfish interest to leave the resulting grand coalition to join any potential alternative one which might feasibly form, and is also man-optimal. Thus, under a realistic assumption set, no player has an interest to “destabilize” the matching, even though he theoretically could. The resulting matching is often better than the naïve Gale-Shapely one for some (not all) of the “boys”, being no worse for the rest. This matching is more realistic (stable) than the one produced by top-trading-cycles method, thus offering a qualitative improvement over the latter. Furthermore, we analyze the situation when players are allowed to make strategic threats (i.e. be willing to sacrifice their own outcome to hurt others), offer a relevant example to illustrate the benefits of this form of play, and ultimately provide an exponential time algorithm which tries to determine a good threat-making strategy. We then briefly examine a few other non-conventional possibilities a player has to affect his outcome. Most common variations to the game model are also described and analyzed with regard to applicability of the methods in this paper. Finally, a few examples of real-life problems which can be modeled and solved with the methods in this paper are presented. 1. OVERVIEW AND MOTIVATION