Econ and math people: solve the Delhi University admission problem

Jun. 26, 2015

Admissions to undergraduate programs in the colleges of Delhi University happen through the so-called ‘cut-off’ system: colleges rank students based on marks in the school-leaving examination and for each program announce a “cut-off” mark. Every student whose score is above the cut-off is eligible to join the program. Since there is a common pool of students applying to different colleges and programs, not all students who are offered admissions in a program join. So the process has to be run in multiple rounds. In each round colleges guess the proportion of those offered admissions who would join. And at the end of each round they must reduce the cut-offs to fill the seats unfilled in the last round. As the cut-offs fall students who get offers from their more preferred programs leave the colleges they had joined in the earlier rounds, creating new vacancies.

It is a testament to the skill of the people running the system that the process currently converges in four to five rounds without too many unfilled vacancies or programs with excess admissions. But in fact the whole drama seems avoidable. In a classic 1962 paper Gale and Shapley considered precisely the problem of matching colleges to applicants and came up with an algorithm that given the ranking of colleges by applicants and the ranking of applicants by colleges comes up with a ‘stable match’, i.e. a match with the property that for every unmatched (student, program) pair either the student does not prefer the program to her current program or the program does not prefer the student to its current students.

Since Gale and Shapley’s paper, there has been a lot of theoretical work on this Stable Marriage Problem as well as many actual implementations of variants of the Gale-Shapley matching algorithm. The 2012 Nobel Memorial Prize in Economics was awarded to Shapley and Alvin Roth for their work in this area.

In a city like Delhi with so many economists and mathematicians, and with the Delhi School of Economics being part of Delhi University, one would expect that the cut-off process could be replaced by the central match that would save both students and colleges much of the trouble.

In fact when I was a student there in the late 90s, Delhi University did run a central match for seats reserved for students from the Scheduled Castes (SC). I’m not sure if it still does. But this central match had a problem.

The students’ choice set for the Delhi University admission problem is very large: there are a few dozen colleges with a few dozen courses each. It is impractical to ask students to rank all possible combinations as required by the Gale-Shapley algorithm. The system run for SC students asked students to list only about a dozen course-college combinations in order of preference. The result was that students who aimed too high would not get selected in any of their choices and lower-ranked colleges would have vacant seats, so the process ended up degenerating into a muliple round one. On the other hand the students who aimed too low would miss out on better opportunities that would be captured by more ambitious students.

To implement Gale-Shapley for Delhi University one would have to address this problem of eliciting preferences. It is not possible to simply assume that students’ preferences are lexicographic over colleges and courses in either order. What could possibly work is a hybrid system that asks for an explicit ranking over, say, twenty choices and then have a low-dimensional backup way of getting preferences over the remaining choices. Even a lexicographic ordering of colleges and programs with the student choosing which to place first might work.

More ambitious might be a statistical (or machine learning approach). There seems to be a large degree of consensus among students on which are the preferred colleges and programs. So by asking a sample of students to give to give a full ranking over all alternatives and then asking the general population of students to only make a carefully chosen set of pairwise comparisons it may be possible to get a good approximation to their preferences. In fact, a web-based system could make the process interactive, with the system guessing a preference over all alternatives and then presenting it to the student to make any alteration they want. That way a poorly performing model or a idiosyncratic student would only lead to some additional work for the student and not a poor match.

I think this is a relatively well-defined problem where econ and math people of Delhi can use their skills to create a great deal of social benefit.