The Instability of Stable Matchings: The Influence of One Strategic Agent on The Matching Market
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Consider a matching problem with n men and n women, with preferences drawn uniformly from the possible (n!)^2n full ranking options. We analyze the influence of one strategic agent on the quality of the other agents' matchings under the Gale--Shapley algorithm. We show that even though the Gale--Shapley algorithm is famous for being optimal for men, one small change in the reported preferences is enough for the women to get a near optimal match. In this case, the quality of the matching dramatically improves from the women's perspective. The expected women's rank is O(^4(n)) and almost surely the average women's rank is O(^2+ϵ(n)) rather than a rank of O(n/(n)) in both cases under a truthful regime.
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