Almost mutually best in matching markets: rank gaps and size of the core

This paper studies the one-to-one two-sided marriage model (Gale and Shapley 1962). If agents’ preferences exhibit mutually best (i.e., each agent is most preferred by her/his most preferred matching partner), there is a unique stable matching without rank gaps (i.e., in each matched pair the agents assign one another the same rank). We study in how far this result is robust for matching markets that are “close” to mutually best. Without a restriction on preference profiles, we find that natural “distances” to mutually best neither bound the size of the core nor the rank gaps at stable matchings. However, for matching markets that satisfy horizontal heterogeneity, “local” distances to mutually best provide bounds for the size of the core and the rank gaps at stable matchings.
The first draft of this paper was written while F. Klijn was LFUI guest professor at Innsbruck University. He gratefully acknowledges the hospitality of Innsbruck University and support from AGAUR-Generalitat de Catalunya (2017-SGR-1359), Ministerio de Ciencia, Innovación y Universidades (ECO2017-88130-P), and the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0563 and CEX2019-000915-S). C. Kah and M. Walzl thank the Austrian Science Foundation (FWF) for support through project P-28632-G27 and SFB63.