So what school should I go to?

The classic question for high school seniors: “So, what are you doing next year? What colleges are you applying to?” Please, give them a break. They get this question way too much, and it only makes them more nervous about their futures. After all, they seem to be under the impression that where they go is a make-or-break issue, and bringing up the subject as if it is important just reinforces that fear.

But while we’re on the topic, where should they apply?

For simplicity, let’s suppose that each college (indexed by a number $i$) has a particular quality level, $q_{i}$, at which every potential student values that college. This can be through the quality of academics, the alumni network, the cost, the location, you name it. One might think that it would be best to apply to as many colleges as possible, since that maximizes your chances of getting in somewhere good. But, like everything in life, there is a cost to doing so. This can be the actual application fee, the time involved in putting together the materials, getting ETS to send your SAT scores, whatever. Let’s fix this cost for each college at $c_{i}$. We can relax these assumptions, but the qualitative result will still be the same.

Suppose there is a large number of people, and we restrict people to applying to one school. Yes, I know, this is an unreasonable assumption, but the qualitative results will again be the same even if we allow for applying to multiple schools; it just makes the math hairier. The probability of getting in is $\frac{n_{i}}{a_{i}}$, where $n_{i}$is the number of slots that the school has, and $a_{i}$ is the number of people who apply.

Let’s consider the Nash equilibrium. Since everyone values each school equally, we will expect that everyone will be indifferent between applying to the various colleges. Thus, we will have, for every college $i, j$,

$\frac{n_{i}}{a_{i}}q_{i}-c_{i}=\frac{n_{j}}{a_{j}}q_{j}-c_{j}$

This can give us the relative admission rates of each school:

$\frac{n_{j}}{a_{j}}=\frac{(n_{i}/a_{i})q_{i}+c_{j}-c_{i}}{q_{j}}$

This equation in and of itself is informative. It shows that the admission rate to a school will be increasing in the cost of applying, all other things being equal. This makes sense: if you make it harder to apply, less people will do so, and this will drive up the admission rate needed by the school to fill all of its slots. Similarly, an increase in the quality of the school drives down the admission rate, since more people will then want to go there, making it more competitive.

So, in summary, what should you do? You should apply to the school which maximizes $\frac{n_{i}}{a_{i}}q_{i}-c_{i}$, which is your expected benefit from applying there. Assuming that everyone else is being rational and doing the same thing, though, then it won’t make much difference where you apply. That being said, this last result will no longer hold if not all people value different schools the same (though the trends for the relative admission rates will still hold), but that makes the analysis too complicated for a mere blog post.

Edit: For a more sophisticated theoretical and empirical model whose basic idea is the same, click here.