Why a Dutch court stopped high school students from swapping schools

Photo above by Roman Mager on Unsplash | images throughout the story made using Piktochart and Freevector

An Amsterdam court ruled in 2015 that students going to high school aren’t allowed to trade places with each other at different schools — at the schools assigned to them by the new lottery system used to match students to schools. Even if both students prefer to trade. Even if they are at each other’s top choice school.

What happened there? Many questions jump out from this surprising little story.

  • How could the new lottery system assign students to each other’s top choice school?
  • Why did the school boards introduce the new lottery system?
  • Why aren’t students allowed to trade places if they want to?
  • Can’t the schools use a lottery system that makes everyone happy?

To answer these questions, we’ll go into the science behind lottery systems for school choice.

(The areas of science that this falls under are social choice theory and market design which are studied by economists, political scientists, mathematicians, philosophers, computer scientists, and psychologists. My research at the University of Amsterdam is about the mathematics and computer science behind social choice methods.)

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.

The two variants of the DA mechanism also work in rounds. But there is an important difference with the Boston mechanism. If a student gets a place in some round, this is only tentative. It’s only penciled in. They might have to give up this place in some future round.

Before the first round, each student submits a numbered list with their top choices (number one school, number two school, etc).

Just like in the Boston mechanism, in the first round every student gets penciled in at their number one choice school. For all schools that have too many students, a lottery decides who gets to stay at the school. Students that don’t get to stay get penciled in at their next best choice for the next round.

In the next round, there might again be schools with too many students penciled in. Again, the lottery decides. Students that were penciled in at these schools in an earlier round might lose their spot, and get penciled in at their next best school. This continues until there is no school with too many students anymore. Only then, the schools’ student lists are written in ink.

If a student doesn’t get into their top choice school in round one, they still get a shot at their number two choice in the next round. Even if it was also already full in the first round. This is important.

The difference between the two variants of the DA mechanism is how the lottery takes place. In the one variant (single-DA), there is one central lottery at the beginning that produces a ranking of all students. Whenever a school has too many students, they keep those students that are highest on this one ranking.

In the other variant (multi-DA), each school has their own lottery at the beginning. These different lotteries (probably) result in different rankings. Each school uses their own ranking to decide what students get to stay in each round.

At first sight, the difference between these two variants seems to be tiny. But this tiny difference has a huge effect! (We’ll get to that.)

Pros and cons

Back to the economists that studied which of the three mechanisms would perform best. They did a survey among Amsterdam students entering high school to find out their preferences. Using this data, they simulated each of the different lottery systems many times.

With the results of the simulations, they were ready to compare the different systems. But this is not so easy. There are several ways to measure how well a mechanism performs.

For example, you can look at how many students (on average) get a place at their top choice school. Single-DA scored better than Boston on this point, but multi-DA scored worse.

You can also look at how many students (on average) get a place in some school in their top three. Both variants of DA scored better on this point than Boston, and multi-DA scored the best of all.

The idea why both variants of DA could give more students a school in their top three than Boston is simple. With the Boston system, if a student loses the lottery at their first choice, the first round is already over. If their number two and three schools are also full after round one, they have no chance of getting into their top three. With the DA mechanisms, students always have a shot at their number two or three schools.

The reason why multi-DA gets more students in their top three than single-DA is also not difficult. With single-DA, students often don’t get into their number one school because they were unlucky in the central lottery. So chances are that they also don’t get in their number two and three schools. With multi-DA, students get a new lot in the lottery at each school. So students are never cursed by a single unlucky lottery.

But it’s impossible to pick a single all-round winner among the different lottery systems. The Boston system also has advantages over the other two.

For instance, most students only need to worry about one choice: what school do they register for in the first round. If they get into this school, they’re done. For the DA systems, every student needs to come up with a numbered list of schools that they like most. The longer the better. This can take a lot of effort, for all students.

The Amsterdam school boards looked at the economists’ results, and they decided to start using multi-DA in 2015. An important argument for this choice was that multi-DA spreads the pain more fairly than the other two options. It won’t give as many students their top choice. But fewer students have to go to their number six or seven school (or worse).

Fairness and strategizing

This argument is about fairness. We can’t make every student entirely happy with a lottery system. But we can try to spread the unhappiness as equally as possible. This is a fairness principle.

Another fairness principle is that all students should have equal chances. It shouldn’t matter how well students (or their parents) understand the lottery system. Or how much money they have. It would be unfair if this would give a student better chances.

This second concept of fairness plays a role when students can strategize. Strategizing means that you submit untrue preferences to get a better outcome.

For both variants of the DA system, this is not possible. It never hurts to give your true preferences. But with the Boston system, strategizing is possible.

Suppose, for example, that a student’s top choice school is hugely popular. But their number two to five schools are only slightly short on places. Then it could make sense to register for the number two school in the first round. They would have good chances to get into the second best school. Going for the number one choice is risky: if they don’t get in, they have no shot at any school in their top five anymore.

The economists found that around 8 percent of Amsterdam students did in fact strategize with the old Boston system. That is, the school where they registered in the first round was not their first choice.

These strategic choices are not always perfect. They depend on what students think that the other students are going to do. It happens that a student doesn’t register at their top choice school in round one, while there are enough places at this school. Their strategic choice backfired!

These mistakes happen around 50 percent of the time when students strategize.

The economists also found that students from poorer neighborhoods make more strategic mistakes. So the possibility of strategizing could give an unfair advantage to richer students.

Now why is trading places forbidden?

So what triggered the court case that I started this article with?

In 2015 the new multi-DA lottery system was used for the first time. Of course, not all students were completely happy with the outcome. (One percent of students didn’t get a place in their top five.)

Unsatisfied students (and their parents) quickly found each other online. They found many cases where two students could trade places, and they would both be better off.

This can happen because each school runs their own lottery. Suppose a student loses the lottery at their number one school (school A). But they win the lottery at school B. And suppose that another student likes school B best. But loses the lottery at school B, and wins the lottery at school A. Then these students won each other’s jackpot. And so they both want to trade.

With these trades, more than 600 further students could get a place at their top choice school, it later turned out. So many students wanted to trade. And nobody would lose with these trades, would they?

But the school boards wouldn’t allow trading.

Then twenty parents took it to court.

And the court ruled in favor of the school boards. Trading remained forbidden.

The judge gave several arguments for this ruling. One argument is about fairness. Before the lotteries take place, every student has the same chances of getting into a school. This is fair.

By trading places, students could get another shot at a place. At schools for which they already had a go in the lottery. This would give them better chances. At the expense of students that don’t (or can’t) trade.

In fact, whenever two students win each other’s jackpot trade, there must always be a third student that’s higher on the ranking at one of the schools — and that wants a spot at this school. A trade between the two bypasses this third student. This is unfair.

Another argument, that is related, is about strategizing. Students were told (correctly) that their best strategy was to give their true preferences. If trading were allowed, this would not be true. It would pay off for students to put popular schools high on their list. A spot at these schools would be good trading material.

So to allow trading would be to enable strategic behavior in future years. This would make the workings of the lottery system less clear. But it could also give an unfair advantage to students that are better at strategizing — and to students that have more resources to make good strategic decisions.

The central point is that efficiency and fairness were in conflict. Allowing trading would make the outcome a bit more efficient: some students would get a better outcome. But it would come at a price. The price is that the system would be less fair.

What can we learn from this?

This story is about more than just a remarkable court case. This is about what we want our society to be like. How can we offer our youth an education that best fits them? How can we divide the limited places at different schools as fairly as possible?

There is no one single best lottery system. They all have advantages and disadvantages. The only way to get every student a place at their number one school is to create more places at these schools.

Unless we make this happen, we have to find a way to share the available spots.

Designing a lottery system to place students at the different schools is not just a mathematical optimization problem. Different design choices can have far-reaching effects. On whether everyone gets equal opportunities, for example.

This is something that we should decide on together, as a society. What system do we want? What is fair? How do we measure efficiency? And so on.

It is important to have an open and informed discussion about these questions. And for this, it should be clear what the consequences of the different options are, and when different principles are in conflict. Like fairness and efficiency in the case of the school lottery in Amsterdam.

The debate about the limited number of places at some schools in Amsterdam is still ongoing.

The year after, in 2016, the Amsterdam school boards switched to the single-DA variant of the mechanism. With this variant, striking situations like those where students get a place at each other’s top choice school can only occur when a student has priority at a school.

Switching to a single lottery instead of one lottery per school didn’t seem important, at first sight. But this small change has the effect that students don’t win each other’s jackpot! Insights like these are why we study social choice methods.