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Letter to the Governments of the EU Member States

In the last few years there has been an intense discussion on the voting procedure in the Council of Ministers of the European Union. With 25 member states (and two more in the near future) it is not a simple task to make reliable judgements on the implications of the various voting systems that have been suggested.
We the undersigned wish to draw the attention of EU Governments to the fact that scientific methods can be used and need to be used to analyse, understand and design complex voting systems. In particular:
From a scientific point of view there are obvious drawbacks to the systems of voting in the European Council discussed so far. The experts on voting theory agree that the Treaty of Nice gives too much power to a number of countries while others obtain less power than appropriate. On the other hand, the draft European Constitution assigns too much power to the biggest and the smallest states in a systematic way, while the middle size countries do not get their due share of influence (see the tables attached). Moreover, the Nice system will be extremely ineffective due to its high quotas.
The 'compromises' proposed recently to change the quota in the draft Constitution either to 65% of the population and 55% of the states or to 55% of the population and 55% of the states make the situation for several countries even worse than in the draft Constitution. As can be shown by mathematical analysis, it is not the quotas that are mainly at fault, but rather the system of proposed weights.
The basic democratic principle that the vote of any citizen of a Member State ought to be worth as much as for any other Member State is strongly violated both in the voting system of the Treaty of Nice and in the rules given in the draft Constitution. It can be proved rigorously that this principle is fulfilled if the influence of each country in the Council is proportional to the square root of its population. This is known as 'Penrose's Square Root Law'. Such a system may be complemented by a simple majority of states.
A voting system that obeys the Square Root Law, i.e., which gives equal power to all citizens, is easily implemented. It is representative, objective, transparent, and effective. Such a system was proposed by Swedish diplomats already in 2000, and recently endorsed in a number of scientific articles (see the list attached).
We urge our politicians to take into consideration the contribution of the scientific community to this issue. We are highly concerned that any system implemented without due regard to the scientific analysis of voting power may become a major drawback to a democratic development in the European Union.
Werner Kirsch
The new qualified majority in the Council of the EU Some comments on the decisions of the Brussels summit
Institut für Mathematik, Ruhr-Universität Bochum, Bochum, Germany.
In Brussels the heads of state or government of the European Union agreed on a new procedure for taking decisions in the EU-Council of Ministers. This procedure is based on the "double majority" as introduced in the draft Constitution proposed by European Convention. In the draft Constitution a qualified majority was defined to consist of at least 50 % of the member states representing at least 60 % of the Union's population.
It is not an easy task to analyze this voting system due to its complexity. In this note we will give a brief analysis of this voting procedure in terms of the mathematical concept of "power indices". The power index, also known as Banzhaf index, measures the share of power a member state has in the EU-Council under a given voting procedure. Power index 100% means total power, 0% no power at all. It seems desirable that any citizen of the EU should have the same influence on decisions of the Council regardless of his or her home country. If we accept this basic democratic rule it is clear that a country with a larger population should have a bigger power. However, a country's power index should not grow proportional to the country's population, but rather to the square root of its population. This means a country four times as big as another one should have twice as much power as the smaller country, not four times as much. This was proven by the British scientist Lionel Penrose as early as 1946. Hence we use the square root law as a benchmark for a fair voting system. In an open letter to the governments of the EU member states a group of almost 50 scientists from ten states called "Scientists for a democratic Europe" suggested a voting system, which obeys the square root law, to the governments of the EU-without success as we shall see below.
The procedure the European Council agreed upon in Brussels contains a number of modifications of the original proposal of the European Convention:
The majorities are increased to 55 % of the states and 65 % of the population.
A proposal is also attained if all but at most three member states agree on it, even if these states represent less than 65 % of the population of the EU.
The majority of states required in 1) must consist of at least 15 states.
This procedure shall be applied from November 2009. According to current plans the EU will consist of (at least) 27 member before November 2009. Since 55% of 27 members are 15 states the rule 3) is most likely unnecessary.
Both in a 25 member Union and in a 27 member Union rule 2) will apply only to very few situations. In a 27 EU there are 227 (about 134 million) ways the member states can vote either "yes" or "no". Rule 2) applies only to 10 such events. The only ways three states represent more than 35 % of the population are: France, UK and Italy,
Or Germany plus one of France, UK and Italy plus one of France, UK, Italy, Spain and Poland. It is therefore clear that this rule will be negligible as the computation of power indices is concerned. The effect of this rule is therefore mainly a psychological one: It expresses the fear that the big states might obtain to much (blocking) power.
One may say that-not for the first time in the history of the Union-the decision rule is unnecessarily complicated.
Below we compute the power indices (Banzhaf indices) for the new rule and compare it to the Nice treaty, the original version of the draft constitution and to Penrose's square root law (SRL) which gives the indices for a system that gives equal power to every citizen in the EU.
Banzhaf indices for various voting procedures in the EU-Council
These figures show that the bigger countries from Germany to Poland lose due to the increases in the required majorities, while all others win. While Germany, France, United Kingdom and Italy are still above the share of power they would obtain from the square root law (SRL), Spain and Poland were already below this share of power with the majorities suggested by the Convention. With the new rule they lose even compared to their previous bad position.
A view on the figures shows that the Nice treaty gave the four biggest states less power than they deserve, especially Germany is treated in an unfair way, while Spain and Poland got too much power. The contrary is true for the procedure laid down by the European Convention: Here, the four biggest states get much more power than they should have, notably Germany. On the other hand Spain and Poland are the big losers both compared to Nice and also compared to the fair share of power (SRL).
Reportedly, the change of the required majorities from 50-60 to 55-65 was meant as a compromise between Germany on one side and Spain and Poland on the other side. Our table of power indices shows that all three countries lose power compared to the 50-60 proposal of the Convention. There is no reason to complain for Germany since this country is still above its fair share of power. However, for Spain and Poland this additional loss of power should be totally unacceptable. These states lose power compared to an already unfair situation.
Presumably the heads of state or government did not realizes this result of their decisions. However, due to the open letter of the "Scientists for a democratic Europe" they could have known.
Bochum, July 1, 2004

Prof. Dr Jesús Mario Bilbao, Uniwersytet w Sewilli, Hiszpania
Dr Frédéric Bobay, ekonomista i pracownik administracji publicznej, Paryż, Francja
Prof. Dr Werner Kirsch, Uniwersytet Ruhry w Bochum, Niemcy
Prof. Dr Moshé Machover, London School of Economics, Wielka Brytania
Prof. Ian McLean, Uniwersytet w Oxfordzie, Wielka Brytania
Dr Bela Plechanovová, Uniwersytet Karola w Pradze, Czechy
Prof. Dr Friedrich Pukelsheim, Uniwersytet w Augsburgu, Niemcy
Dr Wojciech Słomczyński, Uniwersytet Jagielloński, Polska
Prof. dr hab. Karol Życzkowski Uniwersytet Jagielloński i Polska Akademia Nauk, Polska
Pozostali sygnatariusze
Dr Stefan Böcker, WestLB AG, Düsseldorf, Niemcy
Prof. Dr Anne Boutet de Monvel, Uniwersytet Paryż 7, Francja
Prof. Dr David A. Brannan, Uniwersytet Otwarty, Wielka Brytania
Dr Francjasc Carreras, Politechniczny Uniwersytet Katalonii, Barcelona, Hiszpania
Prof. Dr Jean Michel Combes, Uniwersytet Sud Toulon-Var, Francja
Prof. Dr Harrie C. M. de Swart, Uniwersytet w Tilburgu, Holandia
Prof. Dr Peter Eichelsbacher, Uniwersytet Ruhry w Bochum, Niemcy
Dr rer. nat. Martin Fehndrich,, Duisburg, Niemcy
Dr Josep Freixas, Politechniczny Uniwersytet Katalonii, Barcelona, Hiszpania
Dr Uwe Grimm, Uniwersytet Otwarty, Wielka Brytania,
Prof. Dr Fritz Haake, Uniwersytet Duisburg-Essen, Niemcy
Prof. Dr Madeleine O. Hosli, Uniwersytet w Leiden, Holandia
Prof. Dr Alan Huckleberry, Uniwersytet Ruhry w Bochum, Niemcy
Prof. Dr Thomas Jahnke, Uniwersytet w Poczdamie, Niemcy
Prof. Dr Ladislav Kabat, Uniwersytet Comeniusa, Bratysława, Słowacja
Prof. Frédéric Klopp, Uniwersytet Paris-Nord, Francja
Priv.-Doz. Dr Vadim Kostrykin, Instytut Fraunhofera Technologii Laserowej, Aachen, Niemcy
Prof. Dr Dominique Lepelley, Uniwersytet Caen Basse-Normandie, Francja
Dr Antoni Magańa, Uniwersytet w Sewilli, Hiszpania
Dipl.-Math. Stefan Maus, Uniwersytet w Maastricht, Holandia
Dipl.-Wi.-Ing. Oliver Mayer, Uniwersytet w Karlsruhe, Niemcy
Priv.-Doz. Dr Peter Müller, Uniwersytet w Getyndze, Niemcy
Dr Stefan Napel, Uniwersytet w Hamburgu, Niemcy
Priv.-Doz. Dr Norbert Peyerimhoff, Uniwersytet Ruhry w Bochum, Niemcy
Prof. Victoriano Ramírez Gonzáles, Uniwersytet Granady, Hiszpania
Prof. Dr Rudolf A. Roemer, Uniwersytet Warwick, Wielka Brytania
Dr Agnieszka Rusinowska, Szkoła Główna Handlowa w Warszawie, Polska
Prof. Dr Pieter H.M. Ruys, Uniwersytet w Tilburgu, Holandia
Prof. Maurice Salles, Uniwersytet Caen Basse-Normandie, Francja
Prof. M. Remzi Sanver, Uniwersytet Bilgi, Istambuł, Turcja
Dr Bernd Sing, Uniwersytet w Bielefeld, Niemcy
Dr Wojciech Słomczyński, Uniwersytet Jagielloński, Polska
Dr Tadeusz Sozański, Uniwersytet Jagielloński, Polska
Prof. Dr Bernard Steunenberg, Uniwersytet w Leiden, Holandia
Prof. Dr Peter Stollmann, Uniwersytet Techniczny w Chemnitz, Niemcy
Prof. Dr José Ignacio Torreblanca, Narodowy Uniwersytet Edukacji Niestacjonarnej, Hiszpania
Dr Ivan Veselic, California Institute of Technology, USA
Dr Simone Warzel, Uniwersytet Princeton, USA
Prof. Tomasz Zastawniak, Uniwersytet w Hull, Wielka Brytania

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