Penrose method
Encyclopedia
The Penrose method is a method devised in 1946 by Professor Lionel Penrose
for allocating the voting weights of delegations (possibly a single representative) in decision-making bodies proportional to the square root
of the population represented by this delegation. Under certain conditions, this allocation achieves equal voting powers (as defined by the Penrose–Banzhaf index) for all people represented, independent of the size of their constituency, district, or state. A proportional allocation would result in excessive voting powers for the electorates of a larger constituencies.
A precondition for the appropriateness of the method is en bloc voting of the delegations in the decision-making body: A delegation cannot split its votes; rather each delegation has just a single vote to which different weights are applied. Another precondition is that the opinions of the various citizens are not related (statistical independence) within each country. The representativity of each delegation results from statistical fluctuations within the country, and then, according to Penrose, "small electorate are likely to obtain more representative governments than large electorates." A mathematical formulation of this idea results in the square root rule.
Accordingly, the Penrose method has been proposed for apportioning representation in a United Nations Parliamentary Assembly
, and for voting in the Council of the European Union
. Other bodies where the Penrose method would be appropriate include the US Presidential Electoral College and the Bundesrat of Germany
.
Currently, the Penrose method is not used for any notable decision-making body.
The Penrose method became revitalised within the European Union
when it was proposed by Sweden in 2003 amid negotiations on the Amsterdam Treaty
and by Poland June 2007 during summit on the Treaty of Lisbon
. In this context, the method was proposed to compute voting weights of member states in the Council of the European Union.
Currently, the voting in the Council of the EU does not follow the Penrose method. Instead, the rules of the Nice Treaty are effective between 2004 and 2014, under certain conditions until 2017. The associated voting weights are compared in the table to the right along with the population data of the member states.
Besides the voting weight, the voting power (i.e., the Penrose–Banzhaf index) of a member state also depends on the threshold percentage needed to make a decision. Smaller percentages work in favor of larger states. For example, if one state has 30% of the total voting weights while the threshold for decision making is at 29%, this state will have 100% voting power (i.e., an index of 1). For the EU-27, an optimal threshold, at which the voting powers of all citizens in any member state are almost equal, has been computed at about 61.6%. After the university of the authors of this paper, this system is referred to as the "Jagiellonian Compromise". The threshold decreases with the number of the member states.
Under the Penrose method, the relative voting weights of the most populous countries are lower than their proportion of the world population. In the table below, the countries' voting weights are computed as the square root of their year-2005 population in millions. This procedure was originally published by Penrose in 1946 based on pre-World War II
population figures.
In practice, the theoretical possibility of the decisiveness of a single vote is questionable. Elections results that come close to a tie are likely to be legally challenged, as was the case in the US presidential election in Florida in 2000
. After this experience, it appears naive to think that a single vote can be pivotal.
In addition, a minor technical issues is that the theoretical argument for allocation of voting weight is based on the possibility that an individual has a deciding vote in each representative's area. This scenario is only possible when each representative has an odd number of voters in their area.
Lionel Penrose
Lionel Sharples Penrose, FRS was a British psychiatrist, medical geneticist, mathematician and chess theorist, who carried out pioneering work on the genetics of mental retardation. He was educated at the Quaker Leighton Park School and St...
for allocating the voting weights of delegations (possibly a single representative) in decision-making bodies proportional to the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
of the population represented by this delegation. Under certain conditions, this allocation achieves equal voting powers (as defined by the Penrose–Banzhaf index) for all people represented, independent of the size of their constituency, district, or state. A proportional allocation would result in excessive voting powers for the electorates of a larger constituencies.
A precondition for the appropriateness of the method is en bloc voting of the delegations in the decision-making body: A delegation cannot split its votes; rather each delegation has just a single vote to which different weights are applied. Another precondition is that the opinions of the various citizens are not related (statistical independence) within each country. The representativity of each delegation results from statistical fluctuations within the country, and then, according to Penrose, "small electorate are likely to obtain more representative governments than large electorates." A mathematical formulation of this idea results in the square root rule.
Accordingly, the Penrose method has been proposed for apportioning representation in a United Nations Parliamentary Assembly
United Nations Parliamentary Assembly
A United Nations Parliamentary Assembly is a proposed addition to the United Nations System that would allow for participation of member nations' legislators and, eventually, direct election of United Nations parliament members by citizens worldwide...
, and for voting in the Council of the European Union
Voting in the Council of the European Union
The procedures for voting in the Council of the European Union are described in the treaties of the European Union. The Council of the European Union has had its voting procedure amended by subsequent treaties and currently operates on a system brought forth by the Treaty of Nice...
. Other bodies where the Penrose method would be appropriate include the US Presidential Electoral College and the Bundesrat of Germany
Bundesrat of Germany
The German Bundesrat is a legislative body that represents the sixteen Länder of Germany at the federal level...
.
Currently, the Penrose method is not used for any notable decision-making body.
The EU proposal
| Member state | Population | Nice Treaty of Nice The Treaty of Nice was signed by European leaders on 26 February 2001 and came into force on 1 February 2003. It amended the Maastricht Treaty and the Treaty of Rome... |
Penrose | |||
|---|---|---|---|---|---|---|
 Germany |
82.54m | 16.5% | 29 | 8.4% | 9.55% | |
 Early Modern France |
59.64m | 12.9% | 29 | 8.4% | 8.11% | |
| 59.33m | 12.4% | 29 | 8.4% | 8.09% | ||
 Italy |
57.32m | 12.0% | 29 | 8.4% | 7.95% | |
 Spain |
41.55m | 9.0% | 27 | 7.8% | 6.78% | |
 Poland |
38.22m | 7.6% | 27 | 7.8% | 6.49% | |
 Kingdom of Romania |
21.77m | 4.3% | 14 | 4.1% | 4.91% | |
 Netherlands |
16.19m | 3.3% | 13 | 3.8% | 4.22% | |
 Greece |
11.01m | 2.2% | 12 | 3.5% | 3.49% | |
 Portugal |
10.41m | 2.1% | 12 | 3.5% | 3.39% | |
 Belgium |
10.36m | 2.1% | 12 | 3.5% | 3.38% | |
| 10.20m | 2.1% | 12 | 3.5% | 3.35% | ||
 Hungary |
10.14m | 2.0% | 12 | 3.5% | 3.34% | |
 Sweden |
8.94m | 1.9% | 10 | 2.9% | 3.14% | |
 Austria |
8.08m | 1.7% | 10 | 2.9% | 2.98% | |
 Kingdom of Bulgaria |
7.85m | 1.5% | 10 | 2.9% | 2.94% | |
 Denmark |
5.38m | 1.1% | 7 | 2.0% | 2.44% | |
 Slovakia |
5.38m | 1.1% | 7 | 2.0% | 2.44% | |
 Finland |
5.21m | 1.1% | 7 | 2.0% | 2.39% | |
 Republic of Ireland |
3.96m | 0.9% | 7 | 2.0% | 2.09% | |
 Lithuania |
3.46m | 0.7% | 7 | 2.0% | 1.95% | |
 Latvia |
2.33m | 0.5% | 4 | 1.2% | 1.61% | |
 Slovenia |
2.00m | 0.4% | 4 | 1.2% | 1.48% | |
 Estonia |
1.36m | 0.3% | 4 | 1.2% | 1.23% | |
 Cyprus |
0.72m | 0.2% | 4 | 1.2% | 0.89% | |
 Luxembourg |
0.45m | 0.1% | 4 | 1.2% | 0.70% | |
 Malta |
0.40m | 0.1% | 3 | 0.9% | 0.66% | |
 European Union |
484.20m | 100% | 345 | 100% | 100% | |
The Penrose method became revitalised within the European Union
European Union
The European Union is an economic and political union of 27 independent member states which are located primarily in Europe. The EU traces its origins from the European Coal and Steel Community and the European Economic Community , formed by six countries in 1958...
when it was proposed by Sweden in 2003 amid negotiations on the Amsterdam Treaty
Amsterdam Treaty
The Amsterdam Treaty, officially the Treaty of Amsterdam amending the Treaty of the European Union, the Treaties establishing the European Communities and certain related acts, was signed on 2 October 1997, and entered into force on 1 May 1999; it made substantial changes to the Maastricht Treaty,...
and by Poland June 2007 during summit on the Treaty of Lisbon
Treaty of Lisbon
The Treaty of Lisbon of 1668 was a peace treaty between Portugal and Spain, concluded at Lisbon on 13 February 1668, through the mediation of England, in which Spain recognized the sovereignty of Portugal's new ruling dynasty, the House of Braganza....
. In this context, the method was proposed to compute voting weights of member states in the Council of the European Union.
Currently, the voting in the Council of the EU does not follow the Penrose method. Instead, the rules of the Nice Treaty are effective between 2004 and 2014, under certain conditions until 2017. The associated voting weights are compared in the table to the right along with the population data of the member states.
Besides the voting weight, the voting power (i.e., the Penrose–Banzhaf index) of a member state also depends on the threshold percentage needed to make a decision. Smaller percentages work in favor of larger states. For example, if one state has 30% of the total voting weights while the threshold for decision making is at 29%, this state will have 100% voting power (i.e., an index of 1). For the EU-27, an optimal threshold, at which the voting powers of all citizens in any member state are almost equal, has been computed at about 61.6%. After the university of the authors of this paper, this system is referred to as the "Jagiellonian Compromise". The threshold decreases with the number of the member states.
The UN proposal
According to INFUSA, "The square-root method is more than a pragmatic compromise between the extreme methods of world representation unrelated to population size and allocation of national quotas in direct proportion to population size; Penrose showed that in terms of statistical theory the square-root method gives to each voter in the world an equal influence on decision-making in a world assembly".Under the Penrose method, the relative voting weights of the most populous countries are lower than their proportion of the world population. In the table below, the countries' voting weights are computed as the square root of their year-2005 population in millions. This procedure was originally published by Penrose in 1946 based on pre-World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
population figures.
Criticisms
It has been claimed that the Penrose method is restricted to votes for which public opinion is equally divided for and against. A study of various elections has shown that this equally-divided scenario is not typical; these elections suggested that voting weights should be distributed according to the 0.9 power of the number of voters represented (in contrast to the 0.5 power used in the Penrose method).In practice, the theoretical possibility of the decisiveness of a single vote is questionable. Elections results that come close to a tie are likely to be legally challenged, as was the case in the US presidential election in Florida in 2000
United States presidential election in Florida, 2000
The 2000 United States presidential election in Florida took place on November 7, 2000 as it did in the other 49 states and D.C., which was part of the 2000 United States presidential election...
. After this experience, it appears naive to think that a single vote can be pivotal.
In addition, a minor technical issues is that the theoretical argument for allocation of voting weight is based on the possibility that an individual has a deciding vote in each representative's area. This scenario is only possible when each representative has an odd number of voters in their area.
External links
- The Double Majority Voting Rule of the EU Reform Treaty as a Democratic Ideal for an Enlarging Union : an Appraisal Using Voting Power Analysis, D. Leech and H. Aziz, University of Warwick (2007).
- Many more references at the web page of American Mathematical Society here.