A Continuum Model of Stable Matching With Finite Capacities

TL;DR

This paper develops a unified continuum framework for stable matching with finite capacities, introducing a probabilistic model that predicts distributions and provides bounds on match quality, extending previous deterministic models.

Contribution

It proposes a novel continuum model assuming Poisson-distributed interest, which guarantees a unique stable outcome and offers analytical insights into match distributions and bounds.

Findings

The model predicts the distribution of cutoffs accurately in small markets.

Provides bounds on students' average rank in homogeneous settings.

Offers analytical expressions for the number of matches in platform pricing scenarios.

Abstract

This paper introduces a unified framework for stable matching, which nests the traditional definition of stable matching in finite markets and the continuum definition of stable matching from Azevedo and Leshno (2016) as special cases. Within this framework, I identify a novel continuum model, which makes individual-level probabilistic predictions. This new model always has a unique stable outcome, which can be found using an analog of the Deferred Acceptance algorithm. The crucial difference between this model and that of Azevedo and Leshno (2016) is that they assume that the amount of student interest at each school is deterministic, whereas my proposed alternative assumes that it follows a Poisson distribution. As a result, this new model accurately predicts the simulated distribution of cutoffs, even for markets with only ten schools and twenty students. This model generates new insights about the number and quality of matches. When schools are homogeneous, it provides upper and lower bounds on students' average rank, which match results from Ashlagi, Kanoria and Leshno (2017) but apply to more general settings. This model also provides clean analytical expressions for the number of matches in a platform pricing setting considered by Marx and Schummer (2021).