Properties of stable matchings in the popular random-matching-market model have been studied for over 50 years. In a random matching market, each agent has complete preferences drawn uniformly and independently at random. Wilson (1972), Knuth (1976) and Pittel (1989) proved that in balanced random matching markets, the proposers are matched to their $\ln n$th choice on average. In this paper, we consider markets where agents have partial (truncated) preferences, that is, the proposers only rank their top $d$ partners. Despite the long history of the problem, the following fundamental question remained unanswered: \emph{what is the smallest value of $d$ that results in a perfect stable matching with high probability?} In this paper, we answer this question exactly -- we prove that a degree of $\ln^2 n$ is necessary and sufficient. That is, we show that if $d < (1-\epsilon) \ln^2 n$ then no stable matching is perfect and if $d > (1+ \epsilon) \ln^2 n$, then every stable matching is perfect with high probability. This settles a recent conjecture by Kanoria, Min and Qian (2021). We generalize this threshold for unbalanced markets: we consider a matching market with $n$ agents on the shorter side and $n(\alpha+1)$ agents on the longer side. We show that for markets with $\alpha =o(1)$, the sharp threshold characterizing the existence of perfect stable matching occurs when $d$ is $\ln n \cdot \ln \left(\frac{1 + \alpha}{\alpha + (1/n(\alpha+1))} \right)$. Finally, we extend the line of work studying the effect of imbalance on the expected rank of the proposers (termed the ``stark effect of competition''). We establish the regime in unbalanced markets that forces this stark effect to take shape in markets with partial preferences.

翻译：在随机匹配市场模型中，稳定匹配的性质已被研究了超过50年。在随机匹配市场中，每个代理的完全偏好是独立均匀随机生成的。Wilson（1972）、Knuth（1976）和Pittel（1989）证明了在平衡随机匹配市场中，提议方平均匹配到其第$\ln n$个选择。本文考虑了代理具有部分（截断）偏好的市场，即提议方仅对前$d$个合作方进行排序。尽管该问题历史悠久，但以下基本问题仍未得到解答：\emph{使完美稳定匹配以高概率存在的最小$d$值是多少？}本文精确地回答了这一问题——我们证明了$\ln^2 n$的度数是必要且充分的。即，我们表明若$d < (1-\epsilon) \ln^2 n$，则没有稳定匹配是完美的；若$d > (1+ \epsilon) \ln^2 n$，则每个稳定匹配都以高概率完美。这解决了Kanoria、Min和Qian（2021）的最新猜想。我们将这一阈值推广至非对称市场：考虑短边有$n$个代理、长边有$n(\alpha+1)$个代理的匹配市场。我们证明，对于$\alpha =o(1)$的市场，刻画完美稳定匹配存在性的尖锐阈值出现在$d$等于$\ln n \cdot \ln \left(\frac{1 + \alpha}{\alpha + (1/n(\alpha+1))} \right)$时。最后，我们扩展了关于非对称性对提议方期望排名影响（即“竞争的尖锐效应”）的研究。我们确定了在部分偏好市场中迫使该效应出现的非对称市场条件。