### Reverse Optimization and Admissions Markets

The hot topic in lab nowadays is reverse optimization. In normal optimization, we are given information about a system and try to figure out which inputs will make it do its thing at the lowest cost (or highest profit). In reverse optimization, we are given the inputs that we know produce optimal behavior, and we try to figure out the mechanics of the system.

If the system is a market and the optimal inputs are the quantities of supply and demand at equilibrium, then we can use reverse optimization to figure out the fundamental parameters of the market: How much value do people ascribe to each commodity? This is called econometrics.

Here are a few proof-of-concept examples of reverse optimization I have developed for the admissions market I am modeling. The first example concerns student preferences. Given information about the demand (entering class size) for each school and its admissions standards, I show how to derive the probability weight or preferability given to each university when it enters a student’s choice set. This exercise uses real data on SAT scores and admissions rates from a somewhat arbitrary set of 98 universities.

In general, the schools that top the list are popular overall. But it’s surprising to see that, for example, UW Madison has a higher preferability rating than Princeton, which means (by the definition of the preference parameters) that a student admitted to both schools is slightly more likely to choose UW Madison.

This is an artifact of the design of the model, which considers only SAT math scores. UW Madison has unusually high math scores. Perhaps this is because most UW Madison applicants take the ACT (which is more popular in the Midwest) rather than the SAT; hence, the subset of admits who submitted the SAT are out-of-state students who are atypically zealous about their college search.

But for the sake of argument, let’s take the results at face value and try to understand the economic relationship between cutoffs, demand, and what I am calling “true yield.” In the model, Princeton’s cutoff is 0.94, which means 6 percent or 5766 of the 96094 students in the market are able to get into Princeton. The number who chose to attend is 1285, for a true yield of 22 percent.

On the other hand, UW Madison’s cutoff is 0.81, so 19 percent of students can get into UW Madison, and of them 6279 choose to attend, for a true yield of 34 percent.

The inevitable conclusion, according to these results, is that UW Madison is more preferable overall. Consider it this way: UW Madison enrolls five times as many students as Princeton. And every student who is qualified for Princeton is also qualified for UW Madison. So, if every student who got into both schools chose Princeton, then UW Madison would have to admit six times as many students as Princeton to achieve this level of enrollment. But in reality, UW Madison admits about three times as many students as Princeton. This can only be explained by the fact that some (in fact, a plurality of) students who get into both schools choose UW Madison—again, according to this very limited model.

Clearly, it’s painfully unrealistic to suggest that all colleges have the same ranking of their applicants, let alone that all colleges perform this ranking using the SAT math score. So in a separate example, I consider how to express schools’ diverse (but positively correlated) preferences over the set of students. This is also a case of reverse optimization. Given real admissions outcomes for individual students at a certain university, I show how to extract the optimal weighting used in the college’s assessment function. The assessment function has a special form that is both compact enough to enable reverse optimization, and robust enough to express a range of school preferences, including schools’ desire to admit well-rounded students with a combination of strengths.

As you may have guessed, one of the things I’m working on now is a way put these two models together.

### Nonlinear Dynamics in College Admissions

Even under the fairly restrictive market restrictions given in my previous post, we can generate instances of school-choice markets that exhibit chaotic dynamics. Here is an example. I let tâtonnement process run for 8000 iterations and plotted the final 7000 cutoffs. The blue and red points indicate even and odd iterations; the behavior is bounded and quasiperiodic but has no obvious symmetry. Here is the code.