Matching Markets for School Admissions

The American college admissions process is an enigma to most of us. Why did we all get in to Stanford over many of our equally qualified high school companions? Was it our grades? Our application essays? Our recommendations? For those of us who did not use the Fountain Hopper’s advice on how to access these admissions records, we may never know.

Thousands of teenagers apply to college each year. Many, like my younger brother, end up getting accepted only to their last choice school. Some may not get accepted at all. What if some of these rejections were because the colleges thought these candidates would accept another school and not go towards their yield? After all, it is often a lot easier to get into schools with an expressed intent of going there (i.e. the early decision applications).

Gale & L.S. Shapley have proposed a new way to structure college admissions – a matching market. The problem is structured as follows. There are n students, and each college can admit a quota of q applications. Naturally, each college wants to as close to q students as possible in order to have an optimal class size, and each student wants to get into the best college he or she can get into. Therefore, the students and colleges will rank each other according to their preferences. Then, the colleges and students are matched until the pairings are in equilibrium. This equilibrium is not determined in the traditional way. It is determined by determining students’ preferences for colleges most important, and subsequently having each student apply to their colleges in order until they get accepted. Therefore, there is no incentive to muddle rankings, so that even if the students rank popular schools first, their next best pairing wont be impaired by having a lower ranking for the next best school. The paper proves there will always exist such an equilibrium, so that no student or college has any incentive to change.

Of course, as the authors recognize, college admissions today cannot be simplified into a nice problem. There are human elements involved that do not eliminate the uncertainty as to why student x got school y and student z did not. Colleges may also have a hard time ranking tens of thousands of students on multiple dimensions.

However, despite the caveats in the college admissions process, the Gale Shapley Deferred Acceptance Algorithm has gained traction with public elementary and high schools in New York, San Francisco, and Boston, where students are ranked according to fewer dimensions- i.e. exam scores and distance from school. Still problems such as how to allocate two equally qualified students exist, and scholars such as Stanford’s Dr. Itai Ashlagi are striving to determine how best to do so.

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