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.

翻译：我们揭示了去中心化一对一匹配环境中稳定匹配的脆弱性。Roth和Vande Vate（1990）的经典研究表明，通过随机选择的阻塞对依次匹配，简单的去中心化动态过程能够保证收敛至稳定匹配。我们的首个定理表明：在温和条件下，任何不稳定匹配——包括稳定匹配的微小扰动——都可能通过此类动态过程最终收敛至任意稳定匹配。第二个定理则揭示了脆弱性的另一层面：稳定化过程可能耗时极长。即便在存在唯一稳定匹配的市场中（此时动态过程始终收敛至同一匹配），去中心化交互也需要指数级时长才能实现收敛。稳定匹配的微小扰动可能导致市场偏离稳定状态，并使相当比例的不匹配参与者长期处于错配状态。我们的结论适用于广泛动态过程类别。