arXiv 2402.096672024-02-15cs.GTcs.DMcs.DS

具有部分偏好的不平衡随机匹配市场

Unbalanced Random Matching Markets with Partial Preferences

Aditya Potukuchi, Shikha Singh

AI总结：

本文研究了具有部分偏好的随机匹配市场，证明了在均衡市场中产生完美稳定匹配的截断长度阈值精确为 $\ln^2 n$，并将其推广至不平衡市场，同时揭示了不平衡对提议者期望排名的“竞争显著效应”。

摘要：

在流行的随机匹配市场模型中，稳定匹配的性质已被研究了50多年。在随机匹配市场中，每个代理人的偏好是均匀且独立随机抽取的完整偏好。Wilson (1972)、Knuth (1976)和Pittel (1989)证明了在均衡的随机匹配市场中，提议者平均匹配到其第 $\ln n$ 选择。本文考虑代理人具有部分（截断）偏好的市场，即提议者仅对其前 $d$ 个伴侣进行排名。尽管该问题历史悠久，以下基本问题仍未得到解答：导致以高概率产生完美稳定匹配的 $d$ 的最小值是多少？本文精确回答了该问题——证明 $\ln^2 n$ 的度数是必要且充分的。即，若 $d < (1-ε) \ln^2 n$，则无稳定匹配是完美的；若 $d > (1+ ε) \ln^2 n$，则每个稳定匹配都以高概率是完美的。这解决了 Kanoria, Min 和 Qian (2021) 的近期猜想。我们将此阈值推广至不平衡市场：考虑短边有 $n$ 个代理人、长边有 $n(α+1)$ 个代理人的匹配市场。我们证明，对于 $α=o(1)$ 的市场，刻画完美稳定匹配存在性的精确阈值出现在 $d$ 为 $\ln n \cdot \ln \left(\frac{1 + α}{α+ (1/n(α+1))} \right)$ 时。最后，我们扩展了研究不平衡对提议者期望排名影响（称为“竞争的显著效应”）的研究路线。我们在具有部分偏好的市场中确立了不平衡市场中迫使该显著效应形成的状态。

英文摘要：

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 $d < (1-ε) \ln^2 n$ then no stable matching is perfect and if $d > (1+ ε) \ln^2 n$, 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 $n(α+1)$ agents on the longer side. We show that for markets with $α=o(1)$, the sharp threshold characterizing the existence of perfect stable matching occurs when $d$ is $\ln n \cdot \ln \left(\frac{1 + α}{α+ (1/n(α+1))} \right)$. 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.

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