A Better Way to Match Exchange Students and Schools?

Students in the Maastricht University exchange program provide an ordered list of their six most-preferred schools. Despite this, because their preferences matched many other students’ preferences and/or they were far back in the queues, 47 of the 357 participating students have not yet been placed.

Is there any way to do a better job of matching? Does the path-breaking work of Alvin Roth (more here) give any guidelines to improving the efficiency of this complicated matching scheme? One possibility is a two-stage process, asking students who are unplaced after Round 1 to list their preferences over the remaining available list of unfilled places. One could even iterate additional rounds until all students are placed. Both students and those foreign universities that are not attracting enough Maastricht students would be better off. But can one do better than this simple addition to the matching process?


The addition of rounds into student's preferences would surely make less demanded universities better off, and unplaced students better off as well. Another thing that could be of a -little- help would be to ask students to be honest with themselves. Comparing themselves to the best student and see where they fall on the rankings. This would eliminate the best school as their number 1 choice, but would have the second best school as the #1 choice. This would place students into universities at a faster pace, since it will shorten the number of rounds necessary to place all applying students.

Tom Roderick

Perhaps extending the list to 10 most preferred?

Daniel Gross

You might ask students to weight their preferences, but that leaves you with the question of how to interact GPA and strength-of-preferences to set an allocation. Shot in the dark idea: stage 1 assignment of top 50% GPA students to their absolute preferences, thus rewarding scholastic ability; stage 2 assignment of remaining students into remaining slots by weighted preferences, to improve allocative efficiency.

I'm curious to see if Al Roth picks up this story for his blog this weekend.


I suggest using Dijkstra's Guarded Gate. If the student does not get placed into one of their top 6 schools, then randomly place them into a slot in one of the remaining schools. This creates an incentive to make sure to pick at least one "unpopular" school. Eventually, you'll find that no one ever gets placed randomly - enough students will pick a "safety school" for their sixth choice that it becomes unnecessary.


I like Tom's suggestion of giving the students more slots.

I'd also give the students more information about which schools are generally oversubscribed with GPA ranges of the students who are selected for those schools. Maastricht exchange students at UCLA generally have a GPA in the range of 3.5-3.7, exchange students at the University of Minnesota have a range of 3.1-3.4 (I'm making these numbers up, obviously, but I'm guessing Minnesota wouldn't be as popular because it's cold here during the school year). People often make decisions based on name recognition, so give the students more information to make them aware of options and resources to learn about lesser-known options.

My brother's currently in medical school, and he chose the schools he applied to based on which schools usually accept students with his home state, GPA, and MCAT from his undergrad school. He's currently working through the same decision process for residency applications.



Provide students with a complete list of participating schools, and have them order them all from most preferred to least preferred.

Adam Kuczynski

Thumperchica, that would be rather tedious, since there are dozens of universities from which we students can choose. Furthermore, a necessary condition for it to work is that people have completely transitive preferences, which is an axiom that surprisingly often doesn't hold in real life.

Such a tedious procedure would create huge opportunity costs, and transaction costs (it's essentially bargaining over a contract), and thus is probably not be efficient.

No - I quite like the current system of picking 6 universities. People just need to be realistic about which universities they could actually get into. Buenos Aires is really popular this year, so, given the fact that there are 5 places available, if someone's ranked number 200+, (s)he's probably not going to get admitted.. But perhaps my opinion's biased because I'm ranked 13th :)


Extending the ranking from top 6 to top 10 is an obvious shortcut but also not a very interesting solution in this case. In the limit you could ask students to rank all possible schools which eliminates the problem altogether, but takes a lot of time and has costs.

Doing a second round with the unassigned students and open slots is yet another obvious solution that comes with costs.

I would argue that for this set level of costs the current system works the best. Students should be aware how the system works and understand the risk of not being assigned. Those with lower GPAs should add a couple of less popular universities as a backup plan if they really want to an exchange semester.

As for the high number of unassigned students.. some might think "It's either the best school or no exchange term at all". They prefer not being assigned to being assigned to a subpar uni.



I don't understand your objection to the "rank 10 schools instead of 6." Yes, in the limit, this would extend to listing all schools, which would be very costly--which is why it's not what is suggested.

There is a cost-benefit trade-off in the number of schools students rank. The costs are more time spent investigating and listing schools; the benefits are reducing the number of unmatched students.

If there's a high number of unmatched students, then there's a decent chance that expanding the number of ranking slots by some small amount would pass a cost-benefit test.


All students who did not find a university after the first round are put on a waiting list, and can still go. Their options are decreased immensely, and that is why most students try to get their first 6 options according to their assumed possibilities.
I've heard that this semester more students want to go abroad than usual, which could have caused the problem.


Sounds like a stable marriage problem - hospitals in the US use it to place residents. It's based on a ranking by both the hospitals and the residents. http://en.wikipedia.org/wiki/Stable_marriage_problem#Similar_problems


This seems to be an instance of the "Stable marriage problem" (http://en.wikipedia.org/wiki/Stable_marriage_problem), which is solved by the Gale-Shapley algorithm. Some variant of this is already used to match medical students to residency programs, which is almost identical to the problem with exchange students.

Tom Roderick

I agree with Andreas that extending the list from 6 to a slightly higher value is not a theoretically interesting solution. It would, perhaps, solve the question simply without additional inefficiency in resource expenditure by the office in charge.

Perhaps not all student ranked spots need to be filled. I am unclear if under the current rules all 6 spots need to be ranked by the student. Perhaps not attending should also be an option? Buenos Aires or bust?


I think the best idea would be to QUIT WORRYING so much.

It's an exchange term in which students want to absorb the culture around them and meet lots of new people. It should be during this exchange term that students take electives or classes they are very interested in rather than boring requirements.

In my experience with exchange students we spend so much time worrying about picking the right place, not realizing that if we go someplace else it'll turn out very well. Students read about them and talk to others, but the experience is unique to each person, never being exactly what you expected.

I feel students should order their top 20 schools, if #1 doesn't accept, go to #2, then #3, until they are eventually placed.


This is not a new problem.

I presume for the system to work, students are obligated to attend the program to which they are matched. If this is the case, I don't believe that increasing the size of the rank list will solve the problem. There needs to be a second round for unmatched students.

Or there could be something like the residency match scramble for those graduating medical students who fail to match into a residency program.


As there were 396 places and only 357 students, this a problem of incomplete information. The only information available is the allocation of the previous year. So students who are high ranked, choose medium ranked universities as their save choices. Medium ranked students choose low ranked universities as save choices. So in the end there is a gap some where. This gap is due to incomplete information. Students should know what is left for them. Student #1 starts choosing from the 396 places. Student #2 than can choose from the remaining 395 places and so on, until each student received an allocation. That is a very fair method and it would finish the gambling, which each student has to encounter. I assume that this allocation can be done by a clever software, or by meetings in the university.


Students are NOT obliged to accept the allocated parter university. Furthermore, there is a list of remaining schools that students with no allocation can choose from ("a second round").
Finally, at least top ranked students are well aware of the choices made by their fellow students as we exchange an informal list of top preferences.