Many-to-one Matching Markets
Matching Markets apply to real life situations whenever there is a group of “sellers” or seeders and a group of “buyers” or acquirers of the resource the seeder is giving. Matching markets are used to maximize social welfare, and ideally to create a perfect match between sellers and buyers where each node is assigned to another node that they prefer (sellers to buyers). Although the terms seller and buyer are used here, sellers and buyers are not always the case in matching markets. In many cases there aren’t buyers or sellers at all. Furthermore, real life situations aren’t usually as simple as a basic matching market, although they do use this basis of perfect matching to fit more complicated scenarios.
According to Jonathan Levin, a professor and the Dean of Stanford Graduate School of Business, matching markets also extend to things like residency matches for doctors, military postings, dating websites, and rushing for Greek life. One very interesting example of these is the school choice model.
His lectures state that since both the schools and the students have preferences over each other, this brings up the idea of a “many-to-one” model (Levin s57). Because of this, two requirements come up. The first is that the schools and students can not be forced into a choice. The second is that there is no switching that would increase social welfare. This adds an extra layer to the normal matching of only the buyers having preferences but the sellers having no preference for which buyers buy from them.
It follows, then, that there is a slightly new process used to match the schools to the students in this case. First, schools set preferences for which students they want – in real life, this is an acceptance letter to each student from an unshared preference list that they have. Students are then allowed to match to the schools they’ve received acceptances from, and reject others they don’t want. After a rejection happens by the student (or a matching), all the schools take this student off their preference lists and focus on sending new acceptances to students with lower grades. This process happens until no more offers or rejections happen (Levin s35).
The school choice model is just one example of how a normal perfect matching can be extended to fit more situations by extending the basic idea of it. Complex situations of preference can be simplified, and this allows such huge decisions with a very large amount of seeders and acquirers become more efficiently made.
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