College Rankings: The Perfect Matching Problem
Recently, news broke out that the U.S. News rankings took away Columbia University’s No. 2 ranking after Columbia University admitted to falsifying some of their statistics. [1] Now at No. 18, Columbia University’s new ranking led to many news outlets criticizing the U.S. News rankings, saying that these rankings are meaningless and arbitrary. However, I believe that it could be interesting to examine how these rankings came about in the first place.
Previously, a former INFO2040 student examined college rankings through the lens of game theory and how each school competes with one another to maximize their own rankings. [2] Taking a different approach, college rankings could be viewed as a perfect matching problem. We can create a bipartite graph, with colleges being in one group and rankings in the other group. U.S. News rankings and other websites want to find a perfect matching where each university is paired to a unique ranking. However, every university wants the No. 1 ranking, creating a constricted set, containing all the colleges trying to pair with the first ranking. In class, we learned that matching problems with constricted sets could still receive a perfect matching, if we add “prices” or “valuations” to each of the items. We also learned that there will always be market clearing prices that maximize social welfare. Knowing this, we can use something similar to prices to create market clearing factors for the college ranking matching problem. Taking the U.S. News rankings as an example [3], they use factors such as graduation/retention rates (22%), social mobility (5%), graduation rate performance (8%), undergraduate academic reputation (20%), faculty resources (20%), student selectivity (7%), financial resources (10%), average alumni giving rate (3%), and graduate indebtedness (5%). Using all of these combined factors, the university with the highest number will get the ranking of No. 1. After many calculations, these factors will eventually pair every university to a ranking.
Even if we all may not necessarily agree with the factors used, they do give us a perfect matching between a college and its ranking. Different websites use different factors, which give different valuations, which result in different rankings. If you disagree with the factors that the U.S. News rankings used, you can look up other ranking sites (like niche, world university rankings, or even Wikipedia) that will give you other solutions to this perfect matching problem.
Sources:
[1] WSJ on Columbia’s new ranking:
[2] Game Theory Blog Post:
https://blogs.cornell.edu/info2040/2011/10/05/the-game-theory-of-college-rankings/
[3] U.S. News Ranking Factors Breakdown:
https://www.usnews.com/education/best-colleges/articles/ranking-criteria-and-weights