Note on the size of a stable matching

TL;DR

This paper investigates the size of stable matchings in a two-sided market, establishing bounds on the number of matched pairs and characterizing preferences that achieve these bounds.

Contribution

It provides a lower bound on the size of stable matchings and characterizes preferences that attain this bound in a two-sided market.

Findings

The largest stable matching contains at least half of the maximum possible pairs.

The number of matched pairs in any stable matching is bounded below by half of the maximum.

Preferences that attain the bound are fully characterized.

Abstract

Consider a one-to-one two-sided matching market with workers on one side and single-position firms on the other, and suppose that the largest individually rational matching contains $n$ pairs. We show that the number of workers employed and positions filled in every stable matching is bounded from below by $\lceil\frac{n}{2}\rceil$ and we characterise the class of preferences that attain the bound.