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

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. Œe men rank the women similarly. An interesting problem is to use these ranking lists and €nd a stable matching in the sense of Gale and Shapley. Œis problem formulation naturally models preferences in several real world applications; for example, dating sites, room renting/leŠing, ride hailing and labor markets. Two key questions that arise in this seŠing 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 )? Œese questions address conditions under which it is possible to €nd a unique (stable) partner, and the quality of the stable matching in terms of the rank or the proximity of the partner. 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 man/woman is assigned to a point on the k-dimensional hypercube based on their answers to a set of k questions with binary answers (e.g. , like/dislike). We consider two di‚erent metrics on the hypercube: Hamming and Weighted Hamming (in which the answers to some questions carry more weight). Under both metrics, there are exponentially many stable matchings when k = blognc. Œere is a unique stable matching, with high probability, under the Hamming distance when k = Ω(n6), and under the Weighted Hamming distance when k > (2 + ε) logn for some ε > 0. Furthermore, under the Weighted Hamming distance, we show that log(X )/log(n) → −1, as n → ∞, when k > (1 + ε) logn 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 (a proxy for pickup time). Assuming the locations of the passengers and cabs are independent Poisson processes of di‚erent intensities, we derive bounds on the distribution of X in terms of busy periods at a last-come-€rst-served preemptive-resume (LCFS-PR) queue. We also get bounds on E(X ) using combinatorial arguments.