## College Admissions as a Matching Market

http://www.huffingtonpost.com/2011/10/24/colleges-with-the-lowest-_n_1028153.html

As college students, we are all familiar with the college admissions process. In applying for college and deciding which to attend, most of us had to consider first our values of the colleges we wanted to apply to and which ones were willing to accept us. This system can be modeled as a matching market in which colleges are represented as nodes in a column on the left and students/applicants are represented as nodes in a column on the right. There exists an edge between a college node and student/applicant node if the student applies to that college.

The values that each student has for the colleges may differ greatly based on interests, location of college, costs of attending/possible financial aid packages, whether their friends are applying to nearby colleges, available programs offered, among many other factors. If the student doesn’t want to apply to a college, then the student’s value for that college can be set to zero, and there is no edge connecting the two. Since each college has in mind an approximate number of students it wants to accept and generally more students apply than the number that are accepted, especially for top tier schools, it can be expected that constricted sets will form in the bipartite matching market graph described above.

In this model, perfect matching can also be found so that students are matched up with colleges in a way that produces the greatest social welfare; colleges admit the students that are the greatest benefit to them, and students usually tend to choose the college that offers them the greatest value provided that the college accepts them. The procedure is similar to the one used in class to find market-clearing prices, but there are a few differences, which need to be taken into account since this situation is a little more complicated. Instead of a one to one matching (like all the examples seen so far in this class), perfect matching for this graph can consist of multiple students being connected to each college node. Another difference is that in each round instead of actually raising the price students have to pay, college admissions raise the criterion for acceptance. For example, if a college is in the neighboring set of a constricted set, it will be more selective when deciding which students to accept since there is a greater demand for students to attend the college. Above is a link to an article about the newly released list of colleges with the lowest acceptance rates; these colleges will be the ones involved in constricted sets in the bipartite graph of college admissions. It can be expected that they will be more likely to select the students with more accomplishments, involvement in school activities, higher SAT/GPA, and other criterions the college looks for in students.