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The stable matching problem seeks to pair up equal numbers of participants of two types, using preferences from each participant. The pairing must be stable: no pair of participants should prefer each other to their assigned match. In each round of the Gale–Shapley algorithm, unmatched participants of one type propose a match to the next participant on their preference list. Each proposal is accepted if its recipient prefers it to their current match. The resulting procedure is a [[truthful mechanism]] from the point of view of the proposing participants, who receive their most-preferred pairing consistent with stability. In contrast, the recipients of proposals receive their least-preferred pairing. The algorithm can be implemented to run in time quadratic in the number of participants, and [[linear time|linear]] in the size of the input to the algorithm.
The stable matching problem, and the Gale–Shapley algorithm solving it, have widespread real-world applications, including matching American medical students to residencies and French university applicants to schools. For more, see {{slink|Stable marriage problem|Applications}}.
==Background==
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