Fragile Stable Matchings

TL;DR

This paper investigates the fragility of stable matchings in decentralized markets, showing that small perturbations can lead to any stable matching and that convergence to stability can take exponentially long, highlighting inherent vulnerabilities.

Contribution

It proves that unstable matchings can lead to any stable matching under simple dynamics and that convergence times can be exponentially long, revealing deep fragility in decentralized stable matching.

Findings

Unstable matchings can lead to any stable matching via decentralized dynamics.

Convergence to stability can require exponentially long time.

Small perturbations can cause markets to deviate significantly from stability.

Abstract

We show how fragile stable matchings are in a decentralized one-to-one matching setting. The classical work of Roth and Vande Vate (1990) suggests simple decentralized dynamics in which randomly-chosen blocking pairs match successively. Such decentralized interactions guarantee convergence to a stable matching. Our first theorem shows that, under mild conditions, any unstable matching -- including a small perturbation of a stable matching -- can culminate in any stable matching through these dynamics. Our second theorem highlights another aspect of fragility: stabilization may take a long time. Even in markets with a unique stable matching, where the dynamics always converge to the same matching, decentralized interactions can require an exponentially long duration to converge. A small perturbation of a stable matching may lead the market away from stability and involve a sizable proportion of mismatched participants for extended periods. Our results hold for a broad class of dynamics.