Stable Matchings in Metric Spaces: Modeling Real-World Preferences using Proximity

TL;DR

This paper models stable matchings in metric spaces, analyzing uniqueness and distribution of match distances in applications like dating and ride hailing, using probabilistic and geometric methods.

Contribution

It introduces a framework for stable matchings based on metric space preferences and characterizes conditions for uniqueness and distribution in real-world scenarios.

Findings

Exponential number of stable matchings when k = floor(log n) in hypercube models.

High probability of unique stable matching when k = Omega(n^6) for Hamming metric.

Distribution bounds for match distances in ride hailing modeled as Poisson processes.

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 $k = \lfloor\log n\rfloor$. There is a unique stable matching, with high probability, under the Hamming distance when $k = \Omega(n^6)$, and under the Weighted Hamming distance when $k > (2+\epsilon) \log n$ for some $\epsilon>0$. 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.