The proposed changes would benefit Germany, France, Britain and Italy the most.© ImageSource

The European Union is in danger of adopting unfair voting procedures, according to Polish scientists.

They claim that proposed changes to voting rules for the EU Council of Ministers, the union's most senior decision-making body, will give the individual citizens of some countries much greater influence than others.

Karol Życzkowski and Wojciech Słomczyński of the Jagiellonian University in Kraków say the changes will benefit Germany, France, Britain and Italy, and be detrimental to medium-sized countries; Poland will come off worst of all. They propose a different weighting of votes that would make sure individuals across Europe ended up with the same voting power, no matter what size their country was.

At the moment, each country's representative on the council has a certain number of votes, called the country's voting weight. Proposals outlined in the treaty for a European constitution would change those weights. Under the new system, each country's number of votes would be roughly proportional to the size of its population.

Życzkowski and Słomczyński say that is not fair. On the surface, allocating votes in proportion to a country's population size might seem the best way to give every individual citizen an equal influence over the final vote. But because individuals are voting through an elected representative who wields a block vote, this does not work.

Imagine a two-country union in which one country is twice the size of the other. Citizens in the smaller country would be powerless, since those in larger country could always out-vote them.

Root of the matter

In 1946, the geneticist Lionel Penrose (father of physicist Roger Penrose) considered this problem. Countries with larger populations should have more votes because individual citizens in bigger countries have less say in the election of their country's representative. But Penrose showed that increasing the number of votes in proportion to the population size is over-compensating.

He calculated that a citizen's say in such an election diminishes in proportion to the square root of the population size. So the best way to design a fair system is to give each country a number of votes that is proportional to the square root of its population.

The present weightings used by the EU were assigned by the Treaty of Nice in 2001. They were determined rather arbitrarily by long negotiations, without any mathematical basis to them. By good fortune, they turn out to be not vastly different from those corresponding to the Penrose square-root rule.

"They are not too bad", says Życzkowski, "but not very good either." The draft constitution, however, would make the situation much worse.

Power to the people

The researchers demonstrate the effect the changes will have using another of Penrose's suggestions: that a country's "voting power" can be quantified by counting the number of ways a given country can form coalitions with other countries in order to secure a majority. Lawyer John Banzhaf proposed the same idea independently in 1965, so this measure is now called the Banzhaf index.

Życzkowski and Słomczyński calculate that under the new rules, different countries' voting power will change significantly. Germany's Banzhaf index will increase from 8.6 to 13.4, for example, while Poland's will drop from 8.1 to 6.8. The combined voting power of the EU's 'inner core' countries, Germany, France, Belgium and Luxembourg, will more than double under the proposed constitution. These countries have all fought hard for the adoption of the new rules.

Instead, Życzkowski and Słomczyński want the EU to adopt a square root weighting scheme. That would give Germany a Banzhaf index of 10.3, while Poland's would be 7.1.

The researchers also argue that the quota, or size of majority, needed to carry a vote should be altered from the present level of 71% to 62%. This, they calculate, is the value that makes each country's voting weight translate most directly into actual voting power.

Obvious blunders

Moshé Machover of King's College London, who has studied EU voting procedures for many years, agrees with the conclusions, but points out that there is another reason for reducing the amount of agreement required. As the number of member countries increases it gets harder to secure agreements, he says, so it is important to lower the quota accordingly2. If this threshold is left too high, it is hard for the Council to pass any proposals at all.

Indeed, Machover thinks that Życzkowski and Słomczyński's work would be strengthened by giving more attention to the problem of members' power to block bills, as politicians are often more worried about their ability to block bills than to pass them.

So is the EU likely to listen to the researchers' concerns? Machover is not optimistic. These principles have been understood by experts for decades but have rarely made it into practice, whether in Europe or anywhere else. "You can't imagine how many obvious blunders politicians have made because they don't understand the mathematics," he says. 

• References

  1. Życzkowski, K. & Słomczyński, W. Preprint , http://xxx.arxiv.org/abs/cond-mat/0405396, (2004).
  2. Felsenthal, D. S. & Machover, M. Social Choice and Welfare, in press, (2004).