The mathematics of European voting rights

By: by Wolfgang Munchau

What is a fair voting system for the European Union? It looks as though, thanks to Poland, European leaders will be forced to debate this difficult question at their summit this week.

Since the simplified draft treaty is substantively identical to the old and rejected constitution – minus some cosmetics – the voting system proposed is going to be the same one: passage of legislation requires a coalition of countries representing at least 55 per cent of the member states and 65 per cent of the population. The Poles have threatened a veto unless the second of those two numbers is based on the square root of the population size – to reduce Germany’s influence. It sounds arbitrary, but the Poles have a point. Mathematics is on the side of Poland.

 

To an uninitiated observer, this does not appear immediately obvious. Does it not seem fair that the voting power of a country in an international organisation should be proportional to its population size? The answer is no. In fact, it is totally unfair. The reason is that effective voting power in multi-nation settings such as the EU depends not on voting size but on the ability to form winning coalitions. Large countries are better placed than their relative population size would suggest.

The original, six-member Community is a good example of this counter-intuitive idea. Germany, France and Italy each had four votes in the council of ministers, the Netherlands and Belgium had two and Luxembourg one vote. Germany then had more than 100 times the population of Luxembourg, yet only four times the number of votes.

Intuition might suggest that tiny Luxembourg was surely over-represented. In truth, the opposite was the case. The threshold for a majority was set at 12 votes. Since every member except Luxembourg had an even number of votes, Luxembourg was never in a position to cast a make-or-break vote. Despite being numerically over-represented, Luxembourg in effect had zero voting power. That would have been different if, for example, an odd number had been chosen as the threshold.

So how do you measure effective voting power? Lionel Penrose, the British mathematician and psychiatrist who developed a theory of voting power in the 1940s, concluded that votes in international organisations should be based on the square root of the population. This is where the Poles got their idea. In the 1960s, John Banzhaf, a US attorney, established an index to measure a country’s voting power. There are two versions of the Banzhaf index. The absolute Banzhaf index measures the ability of a country to cast the decisive vote in a winning coalition as a proportion of all coalitions in which that country takes part. In the case of the pre-1973 EU, the absolute Banzhaf index for Luxembourg was precisely zero. For Germany it was 24 per cent. Germany, not Luxembourg, was over-represented.

What about the EU today? With 27 members, there are a total of 133m possible coalitions. The economists Richard Baldwin and Mika Widgrén have calculated the Banzhaf indices for each member state, both under the current regime, established by the treaty of Nice and in force since 2004, and the constitution*. The results clearly support the Polish case. Germany’s absolute Banzhaf index shoots up from about 5 per cent to more than 15 per cent (it would have gone up to 30 per cent under the original draft). The trouble is that everyone’s absolute Banzhaf index also goes up, including Poland’s. How could that be?

The reason is that the constitution dramatically improves the probability of legislation being passed. Mathematically, the passage probability can be defined as the ratio of “winning” coalitions to all coalitions. In the 15-member EU, this ratio was 8 per cent (this means that 8 per cent of all possible coalitions produce a Yes vote). Under the Nice rules it has fallen to 3 per cent and will approach zero as the EU expands further. This is why the present voting system needs to be fixed.

The constitutional treaty raises this ratio to 13 per cent. But as the overall passage probability rises, so does a country’s ability to cast a pivotal vote. This explains why the absolute Banzhaf index rises for everybody, including Poland. The Polish problem is that Germany’s influence would be enormous in relative terms.

Is Poland’s square root solution the only alternative? Of course not. EU leaders could, for example, raise the threshold for population size and number of countries from their 55 and 65 per cent respectively or introduce some complicated new formula – perhaps with a square root in it. There is a quite a bit a leeway left without creating Nice-style gridlock. Professors Baldwin and Widgrén propose another simple and effective solution: drop the voting rules of the constitution and just repair the Nice rules by reducing some of the high thresholds.

The Poles have put their finger on an important issue, though their own answer is not as compelling as they think. If and when EU leaders set out to amend the rules, they should heed the lessons of the past. Any new system needs to fulfil two parallel goals: it needs to make the voting system more effective and it needs to be fair. The Nice system is fair and ineffective. The constitution is effective but unfair.

If they get this wrong again, they will be back at the negotiating table not too long from now. But if they get it right, they will have managed to create the one and only substantive change from the original treaty.

*www.cepr.org/pubs/PolicyInsights/PolicyInsight3.pdf

© The Financial Times Limited 2007

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