Game Theory in School Applications
In such a competitive college application environment nowadays, it can be hard to decide what schools to apply to and how to use one’s early decision opportunity. Tracy Tullis writes about a similar process in her New York Times article “How Game Theory Helped Improve New York City’s High School Application Process”. Much like the college application process, the high school application process in New York City requires eighth grade students to spend months touring schools and completing standardized tests before completing an actual application in which the students must choose which schools they want to apply to. Furthermore, the early action/ early decision process for college applications is mirrored in these high school applications because they require students to make a “wish list” of the top 5 high schools they would like to attend. Tullis writes how decades ago, the high school application process created what’s called a congested market — high schools would pick students purely off their application and wish list. However, this results in the very top students getting matched with multiple schools, giving them a choice of where to go, whereas nearly half got no match at all and ended up at schools statistically worse than any on their wish list originally.
To tackle this problem, professor Roth from Stanford, professor Pathak from MIT, and professor Abdulkadiroglu took inspiration from the “stable marriage” problem, where the goal is to pair an unlimited pair of men and women into stable marriages, where a stable marriage is one where “every player’s preference is optimized.”. The algorithm for optimizing this goes as follows: every man proposes to the woman of his choice, and each woman proposed to rejects all but their favorite suitor, but does not tell the man yet. Then, every man who was rejected proposes to his second favorite woman, and again the women reject all but their favorite suitor. This continues until every man has found his favorite woman that will accept him, and every woman has found her favorite man among the ones who proposed to her. This ensures that every player gets their preference. The idea is similar for the college admission process, except in this case, the students are proposing and the high schools and rejecting. The idea is summarized in the picture below taken from the article.
The idea of these stable pairings in the marriage problem as well as this high school application problem is similar to the idea of market control pairings we have recently learned about in class. Just like when three people are choosing from three products they have different enjoyments from each product, which essentially ranks the order of how much they want each product, the students here are ranking which schools they prefer most. However, the added difficulty here is that the schools also choose what students they want to accept, whereas the products never had a preference as to who bought them. In one way, this simplifies the problem because there is no need to introduce an idea like “price” to even out everyone’s enjoyment from the products. Instead, the algorithm used continuously matches the students with the best possible school that would take them.
One final idea that can be seen in the admissions process is a game theory perspective from the side of the student deciding where to apply. Since applying to college doesn’t allow a student to have a ranking of schools, but instead has the early action/decision system that is supposed to help admissions, a different more binary system develops. Say a student wants to apply to schools A, B, and C with acceptance rates 20, 40, and 60% respectively. Each of these schools offers early decision and raises their admissions rate by 10, 15, and 20% respectively. The student would most prefer school A and least prefer school C. Should the student use their early decision to increase their chance at school A, but have decreased chances at schools B and C if they don’t get into A? Or should the student play it safe and use their early decision on school C and try to maximize their chance of getting into a school? Or finally should the student play the middle ground and use their early decision on school B? If the student can quantify how much they want to go to each school, they can create a payoff matrix of sorts to find a Nash equilibrium or mixed equilibrium. Ultimately, in both this college application scenario and the high school application scenario, it is clear to see that game theory plays a big role in how to maximize one’s satisfaction with schooling.