# Poland's square-root-ness

Richard Baldwin, Mika Widgrén

15 June 2007

Poland is arguing strongly for a revision of the EU voting rule that allocates votes to each member according to the square root of their population. What? The square root, as in raised to the power of ½? Yes. As the Financial Times columnist Gideon Rachman put it in his Monday column, the notion that a something as mathematical as a square root “combines obscurity, absurdity and vehemence.” Actually not. People versed in the game theory of voting know that the square-root is almost sacred. It’s called Pensore’s rule and Sweden pushed it in the IGC 2000 negotiations.

This column reviews the logic of the square root for readers with some numeric skills.

Fairness and square-ness

Strange as it may seem, ensuring that the EU’s keystone decision-making body – the Council of Ministers – is such that each EU citizen has equal power – requires just this. Each Council member should have power in Council that is proportional to the square root of her nation’s population. Why?

The basic reason is that Council decision-making is a two-step procedure:

Ø      Citizens elect national governments, and then

Ø      Governments vote in the Council.

This matters.

In her national election, a typical German citizen has less power than a typical Luxembourger. Each group of voters chooses one government but German voters are 160 times more numerous. Thus in national elections, a German voter has much less influence than as a Luxembourg voter. To ensure that each EU citizen is equally powerful in Council decisions, the German Council representative must have more power than the Luxembourg representative. That much is easy to see, but how much more?

A first guess is that in her national election, a German voter is only 1/160th as influential as a Luxembourg voter is in hers. In this case, making EU citizens equipotent in the Council would require that the German Minister is 160 times more power in the Council than the Luxembourg Minister. It seems right – 1/160th as powerful in the national election and 160 times more powerful in the Council. But this is wrong since it misses a subtly that requires some mental gymnastics to comprehend.

In national elections, two things change as the number of voters rises. First, the likelihood of being critical in a particular winning coalition decreases and – as intuition dictates – it declines linearly with the number of voters. Second, the number of winning coalitions increases. Thus, the German has 1/160th the chance that a Luxembourger does of making or breaking a given winning coalition, but for the German this is applied to many more coalitions. Taking this into account one can see that the German voter’s power is less than that of a Luxembourger in their respective national elections, but the figure is not 1/160th as powerful, it is higher. As a consequence, the German Minister’s power in the Council should not be proportional to the German population; it should be less than proportional.1 The precise answer is that for all EU citizens to be equally powerful in the Council, their Ministers should have power in the Council that is proportional to the square root of their national populations. This is called Penrose’s rule. Admittedly, it is not the easiest concept to grasp, but it is correct and has a cherished position in the mathematics of voting systems.

Take two

Poland is not alone. During the IGC 2000, Sweden pushed the square root rule, as we pointed out in some research at the time.

For people trained in game theory and mathematical statistics, the square root is a snap. Take it step by step. (This is from the 2001 CEPR book on the Treaty of Nice entitled Nice Try.)

If everything in the Council of Ministers were decided by an EU-wide referendum, proportional representation would clearly provide each EU citizen with equal power. But decision-making in the EU is a two-step procedure. A typical Frenchman is less likely to be influential than a Dane since each chooses one government but French voters are more numerous. Thus small-nation citizens have a power-edge going into the Council meeting and to even out the power, the votes of big-nation representatives should have more weight in the Council.

But how much more? The formal power measures discussed in our CEPR Policy Insight, yields a simple answer. National power in the Council should increase with the square root of national population. The reason is that power per citizen in national elections declines with the square root of the population, so national power in the Council should increase with square root in order to have a fair system, i.e. a system where each EU citizen is equally powerful in the Council of Ministers.

Where, you may ask, does the square root come from? The answer requires a bit of maths. Consider a randomly selected yes-no issue and suppose that member nations decide their stance on this issue by a referendum; define PN as the probability that a typical citizen’s vote is critical in the referendum outcome. Then the member states vote in the Council. Define Pms as the probability that the member state is critical in the Council vote. A citizen’s probability of being critical is thus PN times Pms and our fairness metric requires this to be equal for all member states.

Pms has nothing to do with the number of voters (proxied by population), but PN falls at the square root of population. This sounds peculiar since most numerate people would think the probability of being critical in a national election decreases in a straight-line relationship with population. But this misses a subtlety. Two things change with the voter headcount. The probability of a typical voter being critical to a particular winning coalition decreases linearly with the headcount, but the number of distinct winning coalitions rises with the number of voters. The probability of being critical therefore falls at less than alinear pace. The mathematics of combinatorics gives us an exact formula assuming a voter’s stance is randomly determined on a randomly selected issue. Taking M as the minimum number of votes in a winning coalition and n as the number of voters, one can use the binomial distribrition to work out the answer. The precise, the formula is complex , but it can be well approximated as the square root of 2/n(Pi), where n is the number of voters and Pi is 3.14 etc. (This approximation is called Stirling’s formula). Hence the square root.

Now you understand why Poland is so adamant about it. They are just trying to be fair. Well, actually the square root serves them pretty well in terms of power, but that is another topic.

1Try a simple example. With a 50% majority rule and one-vote per citizen, there are 4 winning coalitions when there are 3 citizens (A&B, A&C, B&C, A&B&C). With 5 voters there are 11 winning coalitions (A&B&C, A&B&D, A&B&E, A&C&D, A&C&E, A&D&E, B&C&D, B&C&E, A&B&C&D, A&B&C&E, A&B&C&D&E).

Topics:  EU institutions

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