Gale–Shapley algorithm
Generalizations

In their original work on the problem, Gale and Shapley considered a more general form of the stable matching problem, suitable for university and college admission. In this problem, each university or college may have its own quota, a target number of students to admit, and the number of students applying for admission may differ from the sum of the quotas, necessarily causing either some students to remain unmatched or some quotas to remain unfilled. Additionally, preference lists may be incomplete: if a university omits a student from their list, it means they would prefer to leave their quota unfilled than to admit that student, and if a student omits a university from their list, it means they would prefer to remain unadmitted than to go to that university. Nevertheless, it is possible to define stable matchings for this more general problem, to prove that stable matchings always exist, and to apply the same algorithm to find one.

A form of the Gale–Shapley algorithm, performed through a real-world protocol rather than calculated on computers, has been used for coordinating higher education admissions in France since 2018, through the Parcoursup system. In this process, over the course of the summer before the start of school, applicants receive offers of admission, and must choose in each round of the process whether to accept any new offer (and if so turn down any previous offer that they accepted). The method is complicated by additional constraints that make the problem it solves not exactly the stable matching problem. It has the advantage that the students do not need to commit to their preferences at the start of the process, but rather can determine their own preferences as the algorithm progresses, on the basis of head-to-head comparisons between offers that they have received. It is important that this process performs a small number of rounds of proposals, so that it terminates before the start date of the schools, but although high numbers of rounds can occur in theory, they tend not to occur in practice. It has been shown theoretically that, if the Gale–Shapley algorithm needs to be terminated early, after a small number of rounds in which every vacant position makes a new offer, it nevertheless produces matchings that have a high ratio of matched participants to unstable pairs.
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