## In 2011 Geoffrey Grimmett, Professor of Mathematical Statistics at DPMMS, was approached by MEP Andrew Duff to help with a question concerning democracy: find a fair, objective and transparent method for allocating the seats of the European Parliament to Member States. Eight years on, with elections for the European Parliament having recently taken place, the question is still just as relevant.

Up until now, whenever a new country joined the European Union, seats were allocated through political bartering. A principled method would obviously be better, but is harder to agree than you might think. Since, ideally, the citizens in each member state should all be represented equally, it makes sense for the number of seats a country gets to be proportional to the size of its population. The problem is that population sizes vary hugely — compare Germany's nearly 83 million to Malta's 460,300 — so strict proportionality would mean that smaller countries hardly get a foot in the door.

The method should be durable, transparent and impartial to politics. Geoffrey Grimmett

To help with this, larger EU member states have agreed to be generous: they are happy to be slightly under-represented in Parliament so that smaller Member States can be better represented. This show of solidarity is known as *degressive representation*. The Lisbon Treaty of 2009 enshrined it into law. It also stipulates that no country should have fewer than 6 or more than 96 seats, and that the European Parliament should have no more than 751 seats in total.

In response to Duff's approach on behalf of the European Constitutional Affairs Committee (AFCO) in 2011, a committee of mathematicians from around Europe was formed, working to three non-mathematical constraints. "The first was that the system should be durable with respect to changes in size and shape of the EU," says Grimmett, who established and chaired the committee. "The second was that it should be transparent, we should be able to explain it to people. And the third was that the system should be impartial to politics, it shouldn't favour particular groups or nations."

### The Cambridge Compromise

Grimmett's committee reached the unanimous conclusion that seats should be allocated to member states according to a method they called the *Cambridge Compromise*. The idea is straightforward. To begin with, every Member State, no matter how big or small, is given 5 *base seats*. That's extremely degressive because population sizes don't even enter the picture. The remaining seats are then allocated proportionately to population figures: if a state has a population of $p$, then it gets $p/d$ seats, for some divisor $d$ (which we'll explain in a moment). For example, if the divisor $d$ is chosen to be 1 million, then a country with a population of 10 million gets $$5+\frac{10,000,000}{1,000,000}=5+10=15$$ seats.

The problem here is that in general you can't be sure that $p/d$ is a whole number. "You can't give a country, say, 63.3 MEPs," explains Grimmett. "MEPs simply can't be cut up that way. So you need to decide how to round fractions to whole numbers." The mathematicians settled on upward rounding: when $p/d$ isn't a whole number, then go for the nearest whole number that's larger than $p/d.$ It's been shown that this way of rounding gives a slight advantage to smaller member states over larger ones, something that fits well with degressivity.

Since every country gets 5 base seats the Cambridge Compromise ensures that no country ends up with fewer than 6 seats. If a country is allocated too many seats, you overrule the calculation and cap the number of seats at 96. As for the divisor $d$, it is chosen so that the number of seats adds up to the total number of seats you want there to be in parliament.

### The Power Compromise

The Cambridge Compromise seems easy enough, but it comes with a hitch. If you allocate seats using this method, making sure not to exceed the 751 seat total, you find that some Member States lose seats. This is a no-no for many politicians, which is why Grimmett, with his colleagues Friedrich Pukelsheim and Kai-Frederike Oelbermann from Augsburg, Germany, came up with another method called the *Power Compromise*.

This method again starts by giving every member state 5 base seats. But rather than allocating the remaining seats by calculating $p/d$ as above, we calculate $$p^q/d$$ for some exponent $q$. The exponent is less than 1 and is chosen, along with the divisor $d$, so that no country gets more than 96 seats and that the total number of seats is whichever number you want it to be. As before, fractional numbers are rounded upwards.

For example, if we choose $q=0.9$ and $d=1,000,000$, then for a country with a population of 10 million we get $$p^q/d=\frac{10,000,000^{0.9}}{1,000,000}\approx1.99,$$ which is rounded upwards to 2. Taking into account the 5 base seats, the country therefore gets $5+2=7$ seats.

One disadvantage of the Power Compromise is that power functions are harder to explain to people than straight proportionality and are also harder to interpret. Simply dividing the population size by $d=1,000,000$ tells you how many MEPs you need to have one of them per 1 million citizens. But when dividing $10,000,000^{0.9}$ by $d=1,000,000$ it is not quite clear how to interpret the corresponding population unit.

It turns out, however, that the Power Compromise wouldn't cause any country to lose seats compared to the current allocation, as long as the parliament contains at least 723 seats.

In January 2017, with a view to the May 2019 elections, the European Constitutional Affairs Committee (AFCO) hosted a workshop in Brussels at which it solicited advice from invited mathematicians. The Cambridge Compromise and the Power Compromise were two methods presented at the workshop, alongside two variants developed by others: a modified version of the Power Compromise, and a method based on so-called *adjusted quotas*, which avoids seat losses for a parliament bigger than the permitted 751 seats.

### What is going to happen

Maths appears mysterious to some, but politics is more so. After the workshop in January 2017 AFCO entered a lengthy deliberation period. As far as Brexit is concerned, AFCO took a practical approach: assume that the UK won't be part of the EU for the 2019 to 2024 legislative period and decide on a new composition of the European Parliament which excludes the UK. If the UK remains after all, carry on with the current composition, and if it quits within the 2019 to 2024 period, perform a spontaneous switch from old to new.

In the summer of 2017 AFCO indicated what it wanted for the new composition: no Member State should lose seats, degressivity should be respected, and the total number of seats should be between 700 and 710. This disqualified the Power Comprise, as it needs a parliament of at least 723 seats for no seat losses to occur. In response, Grimmett and his colleagues came up with a variant of the Cambridge Compromise which would fit the bill. Other participants of the workshop also suggested alternatives.

In January 2018, after deliberations behind closed doors, AFCO finally announced the new composition — and surprised everyone by not taking recourse to any kind of method or formula. The mathematicians and politicians seeking an objective and transparent method are disappointed. "AFCO was obliged to agree on a process which is objective, fair, durable, and transparent. It has not met this obligation beyond achieving degressivity," Grimmett and Pukelsheim have written. "Parliament missed this opportunity to proceed from the dark ages to an era of enlightenment."

Andrew Duff, who made initial contact with mathematicians at Cambridge, has blamed the "unfamiliarity of most politicians with mathematics" and pressure from Member States that are currently over-privileged in terms of seats for AFCO's failure to adopt a mathematical formula. In June 2018 the European Commission approved the proposed new composition. Those who have been disappointed can only hope that AFCO revisits the subject in good time for the 2024 European election.

### Further information

You can find links to relevant articles, reports and videos on the webpages of Geoffrey Grimmett, or read a 2011 *Plus* magazine article about Grimmett's involvement with the European Parliament elections. Andrew Duff has written about the issue in a discussion paper of the European Policy Centre.