On the diameter of the polytope of the stable marriage with ties

Abstract: The stable marriage problem with ties is a well-studied and interesting problem in game theory. We are given a set of men and a set of women. Each individual has a preference ordering on the opposite group, which can possibly contain ties. A stable marriage is given by a matching between men and women for which there is no blocking pair, i.e., a men and a women who strictly prefer each other to their current partner in the matching. In this paper, we study the diameter of the polytope given by the convex hull of characteristic vectors of stable marriages, in the setting with ties. We prove an upper bound of $\lfloor \frac{n}{3}\rfloor$ on the diameter, where $n$ is the total number of men and women, and give a family of instances for which the bound holds tight. Our result generalizes the bound on the diameter of the standard stable marriage polytope (i.e., the well-known polytope that describes the setting without ties), developed previously in the literature.