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.

中文翻译：

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论有领带的稳定婚姻的多面体的直径

有关系的稳定婚姻问题是博弈论中一个经过充分研究和有趣的问题。我们有一组男性和一组女性。每个人都有一个对相反组的偏好排序，其中可能包含联系。稳定的婚姻是通过男女之间的匹配而产生的，没有阻止对，即在匹配中，男人和女人严格地偏爱彼此而不是他们当前的伴侣。在本文中，我们研究了在有关系的情况下由稳定婚姻特征向量的凸包给出的多面体的直径。我们证明了 $\lfloor \frac{n}{3}\rfloor$ 在直径上的上界，其中 $n$ 是男性和女性的总数，并给出了一个约束严格的实例族。