A Note on Assortativeness Measures

TL;DR

This paper critically examines and corrects the axiomatizations of assortativeness measures in matching, clarifies their valid classes, and extends the odds ratio to multi-type markets.

Contribution

It provides a counterexample to previous axiomatizations, refines the characterization of assortativeness indices, and generalizes the odds ratio to multi-type settings.

Findings

Counterexample invalidates previous axiomatization.

Characterizes the exact class of indices satisfying the axioms.

Extends the odds ratio to multi-type markets.

Abstract

Chiappori et al. (2025) study several indices of assortativeness in matching, including the aggregate likelihood ratio and the odds ratio. We provide a counterexample showing that their axiomatization of the aggregate likelihood ratio is not valid as stated. We identify the exact class of indices characterized by the axioms in Chiappori et al. (2025). We then show that the axiomatization of the aggregate likelihood ratio can be recovered by adding new axioms. In addition, we point out errors in the axiomatizations of other measures in Chiappori et al. (2025). Finally, we offer a generalization of the odds ratio from two-type markets to multi-type markets.