Stable matching in a community consisting of $N$ men and $N$ women is a classical combinatorial problem that has been the subject of intense theoretical and empirical study since its introduction in 1962 in a seminal paper by Gale and Shapley. In this paper, we study the number of stable pairs, that is, the man/woman pairs that appear in some stable matching. We prove that if the preference lists on one side are generated at random using the popularity model of Immorlica and Mahdian, the expected number of stable edges is bounded by $N \ln N + N$, matching the asymptotic value for uniform preference lists. If in addition that popularity model is a geometric distribution, then the number of stable edges is $\mathcal O(N)$ and the incentive to manipulate is limited. If in addition the preference lists on the other side are uniform, then the number of stable edges is asymptotically $N$ up to lower order terms: most participants have a unique stable partner, hence non-manipulability.

中文翻译：

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具有相关随机偏好的双边匹配市场几乎没有稳定对

在由 $N$ 男性和 $N$ 女性组成的社区中的稳定匹配是一个经典的组合问题，自从 Gale 和 Shapley 于 1962 年在一篇开创性论文中提出以来，该问题一直是激烈的理论和实证研究的主题。在本文中，我们研究稳定对的数量，即出现在某些稳定匹配中的男/女对。我们证明，如果使用 Immorlica 和 Mahdian 的流行模型随机生成一侧的偏好列表，则稳定边的预期数量以 $N \ln N + N$ 为界，匹配统一偏好列表的渐近值。此外，如果流行度模型是几何分布，那么稳定边的数量是 $\mathcal O(N)$ 并且操纵的动机是有限的。如果另外一侧的偏好列表是统一的，