Fragile Stable Matchings
<|reference_start|>Fragile Stable Matchings: 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.<|reference_end|>
@article{rudov2024fragile,
    title={Fragile Stable Matchings},
    author={Kirill Rudov},
    journal={arXiv preprint arXiv:2403.12183},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.12183},
    primaryClass={econ.TH cs.GT}
}