March 12, 2007
Suspect Polish maths and a tricky summit
Poland could be about to make life much harder for Angela Merkel, German chancellor, as she tries to give the kiss of life to the EU’s comatose constitutional treaty. Warsaw is threatening to reopen one of the treaty’s most contentious issues: how much power each country should have in the Council of Ministers, the Union’s main legislative body.
"The proposed voting system in the EU constitution mostly hits Poland, according to mathematicians," said Jaroslaw Kaczynski, Poland’s prime minister last week. I don’t know which mathematicians he’s been speaking to, but he’s plain wrong.
The real maths behind the argument are fascinating, because they underscore a traditional Polish suspicion of their western neighbours - a factor which could make a Poland-Germany summit this week on the Baltic coast especially tricky.
Please stay awake, because this is not generally known. Under the new voting system proposed by the constitution, Poland would actually have more power in the Council of Ministers not less.
The so-called double-majority voting system awards votes directly according to a country’s population. In the latest figures I’ve seen for an EU of 27, Poland’s 36.6m people would give it 8.04 per cent of the votes.
Under the current complex system drawn up at the Nice summit of 2000 Poland has 7.83 per cent of the votes. So what is Mr Kaczynski going on about?
The real problem for him is not that Poland loses clout in the EU - it does not - but that bigger countries like Italy, France and Britain would finally get the share of the votes their population merits. The biggest losers are in fact smaller countries like Belgium, Greece and the Czech Republic.
The biggest winner? Germany of course. The Union’s most populous country would see its clout double from 8.41 per cent of the votes to 17 per cent, ending a scandalous situation in which Germany was neutered of its rightful influence as some kind of ongoing war debt.
Mr Kaczynski may not like it, but most people would regard it as only fair that Germany should have twice as many votes as Poland: after all, it has more than twice as many people. Poland has not yet decided whether to push this grievance all the way and try to reopen the text of the constitution, but such a move would be explosive.
Not only would it further damage German-Poland relations, it would end any hope Angela Merkel, German chancellor, might have of salvaging the useful institutional reforms contained in the constitution.
It is not enough to simply calculate the percentage of EU27 population that are Poles and compare this with the Poland’s current 27 votes out of 346 in the Council of Ministers. You also need to take into account the voting threshold which has to be reached and the possibilities that a country such a Poland has to form a blocking minority. The following analysis for example shows the impact (in an EU25) that two different voting thresholds would have on Poland’s real influence.
http://www.ceps.be/Article.php?article_id=360
Figure 4 of the following report compares differences in voting power in an EU27 using Treaty of Nice voting rules and EU Constitution voting rules. The analysis clearly shows Poland would be one of the big losers.
http://www.cepii.fr/anglaisgraph/communications/pdf/2005/16170305/Widgren.pdf
The ideal distribution of voting weights (such as to given each citizen an equal weight) in a body such as the Council of Ministers whose members represents countries of differing populations is a known mathematical problem to which there is an optimum solution first derived 60 years ago by the mathematician L. Penrose. This is to base voting weights on the square root of population. The method proposed in the defunct EU Constitution would have distributed votes in DIRECT proportion to population which would seriously over-represent large states and Germany in particular. The Treaty of Nice distribution is in fact much closer to the Penrose ideal than is the EU Constitution voting rules.
It is often said that the voting rules in the EU Constitution are “uncontroversial”. This should not be the case when they deviate so much from the Penrose optimum and would strongly reinforce the power of already over-mighty big EU states. The unfair voting rules in the EU Constitution should be reason enough on their own to reject the entire Constitution, or any reincarnation of it in which they were to reappear.
Posted by: John | March 12th, 2007 at 10:00 pm | Report this commentThe double majority rule is beneficial to the largest countries (Germany, France, the United Kingdom, and Italy), due to the ‘per capita’ criterion, and to the smallest countries (from Latvia to Malta), for which the condition ‘per state’ plays a key role. Since the largest and the smallest countries gain relative voting power, it is easy to see that this occurs at the expense of all the medium-sized countries (from Spain to Ireland), which from this point of view are handicapped by the Treaty Establishing a Constitution for Europe. A game-theoretical analysis of the rules of voting in the Council of the European Union shows that the double majority system attributes a much smaller relative voting power to Spain and Poland than the earlier system accepted in the Treaty of Nice in 2001. In this way we obtain a mathematical explanation of the political fact that these two countries were the main opponents of the proposed changes to the voting rules. Professors Richard Baldwin (from Switzerland) and Mika Widgren (from Finland) wrote in 2004: “The Constitutional Treaty rules will break the traditional French German power equality making France the junior partner in the Franco-German alliance. Spain and Poland will lose the Near-Big status they won in the Nice Treaty. The biggest losers are the Mediums (about 10 million citizens). The chief winners are the Big-4; Germany alone wins more than the other Big-4 combined.” So Mr. Kaczynski is definitely right!
Posted by: Wojciech Slomczynski | March 14th, 2007 at 1:01 pm | Report this commentSee also:
Posted by: Wojciech Slomczynski | March 14th, 2007 at 1:02 pm | Report this commentScientists for a democratic Europe: Open letter to the governments of the EU member states. 2004.
http://chaos.if.uj.edu.pl/~karol/pdf/OpenLetter.pdf
Algaba, E., Bilbao, J.M. and Fernandez, J.R. (2007), The distribution of power in the European Constitution, European Journal of Operational Research 176: 1752-1766.
http://www.esi2.us.es/~mbilbao/pdffiles/power_ejor.pdf
Baldwin RE, Widgren M. Council Voting in the Constitutional Treaty: Devil in the Details. (CEPS Policy Briefs No. 53; Centre for European Policy Studies: Brussels; 2004)
http://hei.unige.ch/~baldwin/PapersBooks/Devil\_in\_the\_details\_BaldwinWidgren.pdf
Baldwin RE, Widgren M. Winners and Losers Under Various Dual Majority Rules for the EU Council of Ministers. (CEPR Discussion Paper No. 4450; Centre for European Policy Studies: Brussels; 2004)
http://www.cepr.org/pubs/dps/DP4450.asp
Bilbao JM. Voting Power in the European Constitution. Preprint 2004.
Posted by: Wojciech Slomczynski | March 14th, 2007 at 1:03 pm | Report this commenthttp://www.esi2.us.es/~mbilbao/pdffiles/Constitution.pdf
Bobay, F. (2004), Constitution europeenne: redistribution du pouvoir des Etats au Conseil de l’EU, Economie et Provision 163: 101-115.
http://hussonet.free.fr/bobay04.pdf
Felderer B, Paterson I, Silarszky P. Draft Constitution: The Double Majority Implies a Massive Transfer of Power to the Large Member States - Is this Intended? (Short Policy Paper for EU Convention Forum No. 1; Institute for Advanced Studies: Vienna; 2003) http://www.ihs.ac.at/publications/lib/forum1june2003.pdf
Felsenthal DS, Machover M. Analysis of QM Rule Adopted by the EU Inter-Governmental Conference Brussels, 18 June 2004 [online]. LSE Research Online: London; 2004. http://eprints.lse.ac.uk/archive/00000431
Kirsch W. The New Qualified Majority in the Council of the EU. Some Comments on the Decisions of the Brussels Summit. Preprint 2004.
Posted by: Wojciech Slomczynski | March 14th, 2007 at 1:04 pm | Report this commenthttp://www.ruhr-uni-bochum.de/mathphys/politik/eu/Brussels.pdf
Koornwinder T. De stemverhoudingen in de Europese ministerraad. Preprint 2005.
http://staff.science.uva.nl/~thk/art/popular/EUstemmen.pdf
Laruelle A, Widgren M. Is the Allocation of Voting Power Among the EU States Fair? (CEPR Discussion Paper No. 1402; Centre for European Policy Studies: Brussels; 1996)
http://www.cepr.org/pubs/dps/DP1402.asp
Mabille, L. (2003), Essai sur une juste ponderation des voix au Conseil de l’Union europeenne. Preprint; see also: Dubois, N. Ponderation des voix: la preuve par “27”, Liberation 26/11/2003
http://pageperso.aol.fr/lcmabille/
Plechanovova B. Draft Constitution and the Decision-Making Rule for the Council of Ministers of the EU - Looking for Alternative Solution. (European Integration online Papers (EIoP), Vol. 8, No. 12; 2004
Posted by: Wojciech Slomczynski | March 14th, 2007 at 1:04 pm | Report this commenthttp://eiop.or.at/eiop/pdf/2004-012.pdf
Slomczynski W, Zyczkowski K. Voting in the European Union: The Square Root System of Penrose and a Critical Point. Preprint, May 2004.
http://www.arxiv.org/abs/cond-mat/0405396
Slomczynski, W. Zyczkowski K. From a toy model to the double square root voting system, Homo Oeconomicus (2007), in press.
http://chaos.if.uj.edu.pl/~karol/pdf/SZ07b.pdf
Widgren M. Power in the Design of Constitutional Rules. (European Economy Group, University of Madrid Working Papers No. 23; 2003) http://www.ucm.es/info/econeuro/documentos/documentos/dt232003.pdf
Zyczkowski, K., Slomczynski, W. and Zastawniak, T. (2006), Physics for fairer voting, Physics World 19: 35-37.
http://physicsweb.org/articles/world/19/3/6/1
and many others…
Voting rules implemented by various political or economical bodies may be studied with the help of the tools developed for many decades in game theory. A mathematical theory of indirect voting was initiated after World War II by British psychiatrist and mathematician Lionel S. Penrose (1946) in the context of a hypothetical distribution of votes in the United Nations General Assembly. He introduced the concept of a priori voting power, a quantity measuring the ability of a participant of the voting body to influence the decisions taken. This notion may be also used for analysing rules governing the taking of decisions in the Council of the European Union.
It is important to clearly differentiate here between the voting weight of a given country and its potential voting power, the latter reflecting the extent to which it may influence decisions taken by the Council when all possible coalitions between different countries are taken into consideration. To illustrate the difference with a simple example: a shareholder with 51% of stocks of a company has only 51% of all votes (voting weight) at the shareholders assembly, but he takes 100% of the voting power if the assembly votes by a simple majority rule.
Clearly, with 27 member states and complicated voting procedures it is a non-trivial task to analyse the distribution of power in the Council, since one has to consider more than 134 millions of possible coalitions. To quantify the notion mathematicians introduced the concept called power index, which measures the voting power of each member of the voting body. Although the current scientific literature contains several competing definitions of power indices, one often uses the original concept of Penrose. In his approach, the a priori voting power of a country is proportional to the probability that its vote will be decisive in a hypothetical ballot: should this country decide to change its vote, the winning coalition would fail to satisfy the qualified majority condition.
Which voting system is fairer and more accurate? A partial answer to this question was already given by Penrose, who deliberated principles of an ideal representative voting system, in which every citizen of every country has the same potential voting power. First consider direct elections of the government (which nominates the minister voting on behalf of the entire country in the European Council) in a state with population N. It is easy to imagine that an average German citizen has smaller influence on the election of his government than, for example, a citizen of the neighbouring Luxembourg. Making use of the Bernoulli scheme and the Stirling approximation of the binomials, Penrose proved that in such elections the voting power of a single citizen decays as 1 over the square root of N, given that the votes of citizens are uncorrelated. Thus, the system of indirect voting applied to the European Council would be representative in this sense, if the voting power of each country behaved proportionally to the square root of N, so that both factors cancelled out.
Penrose’s square-root law was first proposed as the basis of a voting system for the Council of Ministers in 1998 not by Poles, but by Dr Annick Laruelle of the Université Catholique de Louvain in Belgium and Professor Mika Widgrén of the Turku School of Economics and Business Administration in Finland. Its application to EU decision making has also been investigated by Dan Felsenthal of the University of Haifa in Israel and Moshé Machover of the London School of Economics and many other authors. A similar system was put forward by Swedish diplomats in 2000. Then, Sweden’s prime minister Göran Persson said, “Our formula has the advantage of being easy to understand by public opinion and practical to use in an enlarged Europe…it is transparent, logical and loyal. Maybe that is why it does not please everybody.” Also the former Irish prime minister John Burton also made in March 2004 numerous positive references to voting systems based on Penrose’s law: “Instead of double majority, we could instead put in the Treaty a new, clear and automatic mathematical formula for allocating voting weights. This could allocate Council voting weights on the same basis as the Treaty allocates seats in the European Parliament, namely by degressive proportionality. Degressive proportionality means giving proportionately bigger weight (relative to population) to the smallest state and a slightly lesser relative weight, (relative to population) to the biggest country - with a proportionate distribution of weights for countries in between. Such a formula has been proposed by researchers in the London School of Economics. Their formula would allocate voting weights to countries on the basis of the square root of their population, rather than the number of population itself. Because the square root of one is one, this formula would be degressively proportional and would give proportionately more voting weight to smaller country, but gradually reduce that advantage as countries moved up the population scale. Such a formula would be automatic and simple. It was supported by Sweden in the I.G.C., and the Irish Presidency should look at it again, if they get into difficulty with the double majority formula.”
Finally, Professor Karol Zyczkowski and I proposed in 2004 an alternative compromise solution for the distribution of votes in the Council of the European Union, free of the defects characterising both the Nice and the Constitution vote distribution systems, and based on the Penrose square root law. Our proposed voting system has stimulated considerable interest among experts in voting theory, and has been dubbed the “Jagiellonian compromise” by the media. Prior to the EU summit in Brussels in June 2004, an open letter in support of square-root voting weights in the Council of Ministers that was endorsed by more than 40 scientists in 10 European countries (including many German scientists) was sent to EU institutions and the governments of the member states. The reaction of politicians has been varied, but inevitably depends more on how the Jagiellonian compromise affects an individual country’s share of the vote than on universal criteria such as simplicity and objectivity.
Posted by: Wojciech Slomczynski | March 14th, 2007 at 1:10 pm | Report this commentThanks for the in-depth analysis Wojciech. It is interesting that Sweden proposed a “fair” system based on the Penrose square root law in the past, which makes it all the more surprising that a manifestly unfair method made it into the rejected Constitution. If however the aim of the Constitution was not fairness but rather to reinforce the influence of the “Old Europe” bloc of France, Germany, Belgium and Luxembourg in an extended EU then it makes more sense, as that bloc would have close to a blocking minority under the EU Constitution rules. But quite why anyone else in Europe would want to see a voting system that unfairly exaggerates of the influence of that group is much less clear.
Posted by: John | March 16th, 2007 at 1:35 pm | Report this commentGeorge Parker hits one nail on the head, but misses a few others. He says that “the so-called double majority voting system awards votes directly according to a country’s population.” Correct. The other leg of the voting system, the 55% majority of member states, will in reality never play a role in decision-making. It can only decide an issue, if almost all the small or medium-sized states are standing against almost all the big ones. Such situations hardly ever occur. So, in practice, it would not give the small states any real weight. For all practical purposes, the double majority is not double.
Germany would certainly be a winner. But Germany is not as bad off, under the Nice treaty, as media make us believe. The rule that 62% of the population must support a decision gives Germany alone almost half a blocking minority. It is blocking minorities that make the decisions in the EU. The proportions between the blocking potentials of Germany and Poland are rather 1 to 1½, which is not so unreasonable. And, in this way, three of the big four combined can keep their blocking minority. True, the difference in weight between France, UK and Italy, and Spain and Poland on the other hand, is minimal.
It is true that Poland (and Spain) would have a slightly greater weight under the double majority. But, unlike the big four, they would lose in blocking potential when the threshold is lowered. And the gap between Poland/Spain and the “really” big countries would widen, which would be a blow to their old ambition to be seen as members of that club.
As Mr. Parker points out, the real losers would be the medium-sized and small countries. The large countries’ weight would increase about 60%, and that of the medium/small would decrease as much, or combined 2½ times on average.
What you see as a “fair” distribution of votes is a matter of political taste. There is no “right” or “wrong” solution. Of course, one can argue for weight in proportion to the population. But it is states and government that are sitting on the Council, not populations. So one could equally well argue for the traditional inter-governmental principle, one state - one vote. The present “degresssive proportionality”, with some over-representation of medium-sized and small states, can be seen as a compromise.
In the last fifteen years many political scientists, economists, mathematicians and even physicist have invested an enormous work and ingenuity in finding the “right” mathematical solution according to the Penrose theories. There is some logic in them, and the mathematics is correct. But where they go wrong, is in their assumptions about decision-making in the EU. They miss the point that decision-making is a negotiating process, not a bingo game. What member states want, is not to tip the balance, but to get their national interests taken care of, together with predictable allies. Most of the 134 million theoretically possible coalitions are highly unlikely.
The square root, as in the Polish model, is over-taken by events. The large states wanted more than that at Nice. And a lowered threshold would reduce their blocking potential even further. In fact, the Nice rules are close to the square root, with two or three major exceptions. Spain got about four votes that Germany would have had, if the Nice votes had been redistributed by the square root, Poland got four that would mainly have gone to Romania, and Luxemburg got two extra. Otherwise the difference is plus/minus one vote for eleven countries, and none for the remaining ten.
Posted by: Axel Moberg | March 20th, 2007 at 7:34 am | Report this commentI see old friends on the Nice / double voting debate around here! Hi to everybody. What I have not really seen is “new” arguments from the Polish side. We could get back to the technical discussion, but I think that we would be missing the point. If back in 2002-2004, the square root law had scant chances of making it through, now it will be impossible. You cannnot simply threat 26 governments with a veto unless you get what you want because then you face 26 vetos against your one veto. It is fact of life, as Spain discovered back in 2004, that being “right” means two things: one, being right; second, 26 governments conceding that you are right.
And by the way, the first proposal to apply the square root law to EU voting was published in European Integration online Papers (EIoP) Vol. 1 (1997) N° 1; http://eiop.or.at/eiop/texte/1997-001a.htm.
Posted by: Jose Ignacio Torreblanca | April 24th, 2007 at 10:15 am | Report this commentWrong link sorry:
http://eiop.or.at/eiop/texte/1997-001a.htm
Old ‘foundations’ and new ‘rules’ - For an enlarged European Union (P.C. Schmitter / J.I. Torreblanca)
Posted by: Jose Ignacio Torreblanca | April 24th, 2007 at 10:18 am | Report this commentHow can George Parker do any calculations if he cannot collect proper data? Poland’s population is 38.6 million and not 36.6 mln as he says. Plus, I think that Germany’s real population is 75 million and not 82,5. Why 7 million people residing in Germany who are denied German citizenship and who are deprived of the right to vote in federal elections should suddenly work to the advantage of Germany. Why thank to them Germany should enjoy greater power in the UE if in Germany they are denied political rights? Germany does not want to give them political rights and citizenship, even if they are second or third generation permanent residents. Germany should be actually punished by the UE for treatment like that.
Posted by: Frank | June 18th, 2007 at 10:43 am | Report this comment