School choice

Why is a lottery system used in the first place? There are enough places (10.000+) available in Amsterdam for all the (ca. 8.000) students entering high school each year.

So there is a spot for everyone. But not all schools are the same. Some schools offer music and dance classes, others offer more sports classes. Some schools have mostly students with richer and higher-educated parents, others have mostly students from disadvantaged backgrounds. Some school might be around the corner for a student, another school might be an hour’s bike ride away.

Students (and their parents) feel differently about what school they would like to go to most. Some schools are very popular, and these don’t have enough places for all students that want to go there. Schools have a limited number of classrooms and teachers. Unless this changes, there’s no way that everyone gets their top choice.

To decide who gets their first choice and who doesn’t, the school boards in Amsterdam use a lottery system. Students submit their preferences, and the lottery system determines who gets to go to which school.

Why use a new lottery system?

So a lottery is used to decide who gets to go to their first choice school. And then who (among the remaining students) gets to go to their number two choice. And then for the third choice. And so on.

There are several different ways to do this. The system that had been used in Amsterdam before 2015 is known as the Boston mechanism. (It’s named after the city of Boston, where it was used at some point.) I’ll explain how this system works in a minute.

Another mechanism (with two variants) that can be used for this is the so-called Deferred Acceptance (DA) algorithm. The two variants are called DA with single tie-breaking (STB) and DA with multiple tie-breaking (MTB) — let’s call them single-DA and multi-DA, for short. These systems are based on insights in the work of Economics Nobel Prize winners Alvin Roth and Lloyd Shapley.

When this Nobel Prize research got attention in the Dutch media in 2012, the Amsterdam school boards invited several economists to study if the DA mechanism could perform better than the Boston mechanism.

Before I turn to their findings, let’s have a closer look at the three lottery systems (Boston, single-DA, and multi-DA) — and at their differences.

Three lottery systems

The Boston mechanism works in rounds. In the first round, every student registers at one school of their choice. If a school has more places than registered students, all students get to go to this school. If a school doesn’t have enough places, a lottery decides who gets in and who doesn’t.

(To keep things simple, we’ll forget about students having priority at a school, for example because they have a sibling at this school already.)

If a student gets into a school in round one, they have a spot at this school. No matter what happens in future rounds.

In the second round, all students that have no school yet get to register at a school that has places left. Again, if a school has enough places remaining for the students that want to go there, they all get in. Otherwise, there is a lottery among the newly registered students for these remaining places.

And so on, until all students have a school. No matter how many rounds it takes.

The important point is that if a student gets a spot at a school in some round, nothing in the next rounds can take their spot away. And the other way around, if a school is full after some round, no student can get a place at this school in the next rounds.