The deferred acceptance algorithm introduced by Gale and Shapley is a centralized algorithm, where a social planner solicits the preferences from two sides of a market and generates a stable matching. On the other hand, the algorithm proposed by Knuth is a decentralized algorithm. In this article, we discuss conditions leading to the convergence of Knuth’s decentralized algorithm. In particular, we show that Knuth’s decentralized algorithm converges to a stable matching if either the Sequential Preference Condition (SPC) holds or if the market admits no cycle. In fact, acyclicity turns out to be a special case of SPC. We then consider markets where agents may prefer to remain single rather than being matched with someone. We introduce a generalized version of SPC for such markets. Under this notion of generalized SPC, we show that the market admits a unique stable matching, and that Knuth’s decentralized algorithm converges. The generalized SPC seems to be the most general condition available in the literature for uniqueness in two-sided matching markets.
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We wish to thank Prof. P.G. Babu, IGIDR, Mumbai, India for his useful comments during the early stage of work. The present version greatly benefited from the comments and suggestions of the associate editor and the two reviewers and we thank them profusely.
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Ramani, V., Rao, K.S.M. Paths to stability and uniqueness in two-sided matching markets. Int J Game Theory 47, 1137–1150 (2018). https://doi.org/10.1007/s00182-017-0603-9