The Law of Large Numbers for Large Stable Matchings

TL;DR

This paper develops a concentration inequality for empirical matching probabilities in large stable matching markets, enabling rigorous statistical inference and proving consistency of estimators under certain preference correlation assumptions.

Contribution

It introduces a new concentration inequality for large stable matchings that accounts for correlated preferences, advancing empirical analysis methods in matching markets.

Findings

Establishes a concentration inequality for empirical matching probabilities.

Proves laws of large numbers for matching statistics.

Demonstrates estimator consistency for matching probabilities and assortative matching.

Abstract

In many empirical studies of a large two-sided matching market (such as in a college admissions problem), the researcher performs statistical inference under the assumption that they observe a random sample from a large matching market. In this paper, we consider a setting in which the researcher observes either all or a nontrivial fraction of outcomes from a stable matching. We establish a concentration inequality for empirical matching probabilities assuming strong correlation among the colleges' preferences while allowing students' preferences to be fully heterogeneous. Our concentration inequality yields laws of large numbers for the empirical matching probabilities and other statistics commonly used in empirical analyses of a large matching market. To illustrate the usefulness of our concentration inequality, we prove consistency for estimators of conditional matching probabilities and measures of positive assortative matching.