We study the stable marriage problem in two-sided markets with randomly generated preferences. We consider agents on each side divided into a constant number of "soft tiers", which intuitively indicate the quality of the agent. Specifically, every agent within a tier has the same public score, and agents on each side have preferences independently generated proportionally to the public scores of the other side. We compute the expected average rank which agents in each tier have for their partners in the men-optimal stable matching, and prove concentration results for the average rank in asymptotically large markets. Furthermore, we show that despite having a significant effect on ranks, public scores do not strongly influence the probability of an agent matching to a given tier of the other side. This generalizes results of [Pittel 1989] which correspond to uniform preferences. The results quantitatively demonstrate the effect of competition due to the heterogeneous attractiveness of agents in the market, and we give the first explicit calculations of rank beyond uniform markets.

中文翻译：

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分层随机匹配市场：排名与受欢迎程度成正比

我们研究了具有随机生成偏好的双边市场中的稳定婚姻问题。我们考虑将每一侧的代理划分为固定数量的“软层”，这直观地表明了代理的质量。具体来说，一层内的每个代理都有相同的公共分数，每一方的代理都有与另一方的公共分数成比例地独立生成的偏好。我们计算在男性最优稳定匹配中每一层的代理对其合作伙伴的预期平均排名，并证明渐近大市场中平均排名的集中度结果。此外，我们表明，尽管对排名有显着影响，但公共分数不会强烈影响代理与另一方的给定层匹配的概率。这概括了 [Pittel 1989] 对应于统一偏好的结果。结果定量地证明了由于市场中代理的异质吸引力引起的竞争影响，我们首次给出了统一市场之外的排名的明确计算。