Stable Matchings in Metric Spaces: Modeling Real-World Preferences using Proximity
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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 = n. 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) 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.
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