[论文解读] Stable Matchings in Metric Spaces: Modeling Real-World Preferences using Proximity

Hossein Karkeh Abadi, Balaji Prabhakar|arXiv (Cornell University)|Oct 15, 2017

Game Theory and Voting Systems参考文献 14被引用 6

一句话总结

本文研究了在度量空间中基于邻近性（例如k维超立方体或直线上的距离）生成偏好的稳定匹配问题，建模了约会网站和叫车服务等现实世界系统。研究发现，当汉明距离或加权汉明距离的维度足够高（k = Ω(n⁶) 或 k > (2+ε)log n）时，稳定匹配以高概率唯一，且期望到稳定匹配伙伴的距离随n呈对数衰减。

ABSTRACT

Suppose each of $n$ men and $n$ women is located at a point in a metric space. A woman ranks the men in order of their distance to her from closest to farthest, breaking ties at random. The men rank the women similarly. An interesting problem is to use these ranking lists and find a stable matching in the sense of Gale and Shapley. This problem formulation naturally models preferences in several real world applications; for example, dating sites, room renting/letting, ride hailing and labor markets. Two key questions that arise in this setting are: (a) When is the stable matching unique without resorting to tie breaks? (b) If $X$ is the distance between a randomly chosen stable pair, what is the distribution of $X$ and what is $E(X)$? We study dating sites and ride hailing as prototypical examples of stable matchings in discrete and continuous metric spaces, respectively. In the dating site model, each person is assigned to a point on the $k$-dimensional hypercube based on their answers to a set of binary $k$ questions. We consider two different metrics on the hypercube: Hamming and Weighted Hamming. Under both metrics, there are exponentially many stable matchings when $k = \lfloor\log n floor$. There is a unique stable matching, with high probability, under the Hamming distance when $k = Ω(n^6)$, and under the Weighted Hamming distance when $k > (2+ε) \log n$ for some $ε>0$. In the ride hailing model, passengers and cabs are modeled as points on the line and matched based on Euclidean distance. Assuming the locations of the passengers and cabs are independent Poisson processes of different intensities, we derive bounds on the distribution of $X$ in terms of busy periods at a last-come-first-served preemptive-resume (LCFS-PR) queue.

研究动机与目标

• 建模现实世界匹配系统（如约会网站和叫车服务）并基于度量空间中的邻近性进行分析。
• 确定在无需打破平局的情况下，稳定匹配唯一的条件。
• 分析此类基于邻近性的模型中稳定配对的分布及期望距离E(X)。
• 推导稳定伙伴数量的概率界以及匹配唯一性的结果。

提出的方法

• 将参与者建模为度量空间中的点（例如，使用汉明距离或加权汉明距离的k维超立方体）。
• 基于参与者到其他个体的距离递增顺序分配偏好，平局时采用随机打破。
• 使用概率分析和集中不等式（如切尔诺夫不等式）来限制平局和多个稳定伙伴的概率。
• 应用并集界和组合论证，证明随着k增大，非唯一匹配的概率趋于零。
• 通过泊松过程和LCFS-PR队列的忙碌期分析叫车模型，以界定X（到稳定伙伴的距离）的分布。
• 在加权汉明距离下，当k > (1+ε)log n时，推导出log(X)/log(n) → -1（当n → ∞）的渐近结果。

实验结果

研究问题

• RQ1 在何种度量空间和维度条件下，稳定匹配无需平局打破即唯一？
• RQ2 在基于邻近性的匹配模型中，随机选择的稳定配对之间的距离X的分布是什么？
• RQ3 此类模型中，期望距离E(X)如何随n变化？
• RQ4 参与者拥有多个稳定伙伴的概率是多少，且该概率如何随维度k的增加而衰减？
• RQ5 由于邻近性导致的偏好相关性如何影响稳定匹配的数量和结构？

主要发现

• 当k = Ω(n⁶)时，汉明距离度量下稳定匹配以高概率唯一。
• 在加权汉明距离下，当k > (2+ε)log n（对任意ε > 0）时，稳定匹配以高概率唯一。
• 在加权汉明距离下，当k > (1+ε)log n时，有log(X)/log(n) → -1（当n → ∞），表明到稳定伙伴的期望距离按n的幂次衰减。
• 在汉明距离下，当k = Ω(n⁶)时，任一参与者拥有多个稳定伙伴的概率随n增加而趋于零。
• 在泊松分布的出租车与乘客的叫车模型中，利用组合方法和排队论方法推导出E(X)的界。
• 多重稳定匹配的概率上界为n²·2⁻ᵏ，当k > (2+ε)log n时，该界趋于零，从而确保以高概率唯一匹配。