# Penrose method

The Penrose method (or square-root method) is a method devised in 1946 by Professor Lionel Penrose[1] 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. This is justified by the fact, that due to the square root law of Penrose, the a priori voting power (as defined by the Penrose–Banzhaf index) of a member of a voting body is inversely proportional to the square root of its size. Under certain conditions, this allocation achieves equal voting powers for all people represented, independent of the size of their constituency. Proportional allocation would result in excessive voting powers for the electorates of 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 weights are applied proportional to the square root of the population they represent. Another precondition is that the opinions of the people represented are statistically independent. 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.

The Penrose method is not currently being used for any notable decision-making body, but it has been proposed for apportioning representation in a United Nations Parliamentary Assembly,[1][2] and for voting in the Council of the European Union.[3][4] Other bodies where the Penrose method could be appropriate include the US Presidential Electoral College and the Bundesrat of Germany.[citation needed]

## The EU proposal

 Member state Population Nice Penrose[3] Germany 82.54m 16.5% 29 8.4% 9.55% France 59.64m 12.9% 29 8.4% 8.11% UK 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% 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% Czech Rep. 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% 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% 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% EU 484.20m 100% 345 100% 100%

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 adjacent table 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%.[3] After the university of the authors of this paper, this system is referred to as the "Jagiellonian Compromise". Optimal threshold decreases with the number ${\displaystyle M}$ of the member states as ${\displaystyle 1/2+1/{\sqrt {\pi M}}}$.[6]

## 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".[2]

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.[1]

 Populationas of 2005 Percent ofworld population Voting weight Percent oftotal weight World 6,434,577,575 100.00% 721.32 100.00% Rank Country 1 People's Republic of China 1,306,313,812 20.30% 36.14 5.01% 2 India 1,080,264,388 16.79% 32.87 4.56% 3 United States of America 297,200,000 4.62% 17.24 2.39% 4 Indonesia 241,973,879 3.76% 15.56 2.16% 5 Brazil 186,112,794 2.89% 13.64 1.89% 6 Pakistan 162,419,946 2.52% 12.74 1.77% 7 Bangladesh 144,319,628 2.24% 12.01 1.67% 8 Russia 143,420,309 2.23% 11.98 1.66% 9 Nigeria 128,771,988 2.00% 11.35 1.57% 10 Japan 127,417,244 1.98% 11.29 1.56% 11 Mexico 106,202,903 1.65% 10.31 1.43% 12 Philippines 87,857,473 1.37% 9.37 1.30% 13 Vietnam 83,535,576 1.30% 9.14 1.27% 14 Germany 82,468,000 1.28% 9.08 1.26% 15 Egypt 77,505,756 1.20% 8.80 1.22% 16 Ethiopia 73,053,286 1.14% 8.55 1.18% 17 Turkey 69,660,559 1.08% 8.35 1.16% 18 Iran 68,017,860 1.06% 8.25 1.14% 19 Thailand 65,444,371 1.02% 8.09 1.12% 20 France 60,656,178 0.94% 7.79 1.08% 21 United Kingdom 60,441,457 0.94% 7.77 1.08% 22 Democratic Republic of the Congo 60,085,804 0.93% 7.75 1.07% 23 Italy 58,103,033 0.90% 7.62 1.06% 24 South Korea 48,422,644 0.75% 6.96 0.96% 25 Ukraine 47,425,336 0.74% 6.89 0.95% 26 South Africa 44,344,136 0.69% 6.66 0.92% 27 Spain 43,209,511 0.67% 6.57 0.91% 28 Colombia 42,954,279 0.67% 6.55 0.91% 29 Myanmar 42,909,464 0.67% 6.55 0.91% 30 Sudan 40,187,486 0.62% 6.34 0.88% 31 Argentina 39,537,943 0.61% 6.29 0.87% 32 Poland 38,635,144 0.60% 6.22 0.86% 33 Tanzania 36,766,356 0.57% 6.06 0.84% 34 Kenya 33,829,590 0.53% 5.82 0.81% 35 Canada 32,400,000 0.50% 5.69 0.79% 36 Morocco 32,725,847 0.51% 5.72 0.79% 37 Algeria 32,531,853 0.51% 5.70 0.79% 38 Afghanistan 29,928,987 0.47% 5.47 0.76% 39 Peru 27,925,628 0.43% 5.28 0.73% 40 Nepal 27,676,547 0.43% 5.26 0.73% 41 Uganda 27,269,482 0.42% 5.22 0.72% 42 Uzbekistan 26,851,195 0.42% 5.18 0.72% 43 Saudi Arabia 26,417,599 0.41% 5.14 0.71% 44 Malaysia 26,207,102 0.41% 5.12 0.71% 45 Iraq 26,074,906 0.41% 5.11 0.71% 46 Venezuela 25,375,281 0.39% 5.04 0.70% 47 North Korea 22,912,177 0.36% 4.79 0.66% 48 Republic of China 22,894,384 0.36% 4.78 0.66% 49 Romania 22,329,977 0.35% 4.73 0.66% 50 Ghana 21,029,853 0.33% 4.59 0.64% 51 Yemen 20,727,063 0.32% 4.55 0.63% 52 Australia 20,229,800 0.31% 4.50 0.62% 53 Sri Lanka 20,064,776 0.31% 4.48 0.62% 54 Mozambique 19,406,703 0.30% 4.41 0.61% 55 Syria 18,448,752 0.29% 4.30 0.60% 56 Madagascar 18,040,341 0.28% 4.25 0.59% 57 Côte d'Ivoire 17,298,040 0.27% 4.16 0.58% 58 Netherlands 16,407,491 0.25% 4.05 0.56% 59 Cameroon 16,380,005 0.25% 4.05 0.56% 60 Chile 16,267,278 0.25% 4.03 0.56% 61 Kazakhstan 15,185,844 0.24% 3.90 0.54% 62 Guatemala 14,655,189 0.23% 3.83 0.53% 63 Burkina Faso 13,925,313 0.22% 3.73 0.52% 64 Cambodia 13,607,069 0.21% 3.69 0.51% 65 Ecuador 13,363,593 0.21% 3.66 0.51% 66 Zimbabwe 12,746,990 0.20% 3.57 0.49% 67 Mali 12,291,529 0.19% 3.51 0.49% 68 Malawi 12,158,924 0.19% 3.49 0.48% 69 Niger 11,665,937 0.18% 3.42 0.47% 70 Cuba 11,346,670 0.18% 3.37 0.47% 71 Zambia 11,261,795 0.18% 3.36 0.47% 72 Angola 11,190,786 0.17% 3.35 0.46% 73 Senegal 11,126,832 0.17% 3.34 0.46% 74 Serbia and Montenegro 10,829,175 0.17% 3.29 0.46% 75 Greece 10,668,354 0.17% 3.27 0.45% 76 Portugal 10,566,212 0.16% 3.25 0.45% 77 Belgium 10,364,388 0.16% 3.22 0.45% 78 Belarus 10,300,483 0.16% 3.21 0.44% 79 Czech Republic 10,241,138 0.16% 3.20 0.44% 80 Hungary 10,081,000 0.16% 3.18 0.44% 81 Tunisia 10,074,951 0.16% 3.17 0.44% 82 Chad 9,826,419 0.15% 3.13 0.43% 83 Guinea 9,467,866 0.15% 3.08 0.43% 84 Sweden 9,001,774 0.14% 3.00 0.42% 85 Dominican Republic 8,950,034 0.14% 2.99 0.41% 86 Bolivia 8,857,870 0.14% 2.98 0.41% 87 Somalia 8,591,629 0.13% 2.93 0.41% 88 Rwanda 8,440,820 0.13% 2.91 0.40% 89 Austria 8,184,691 0.13% 2.86 0.40% 90 Haiti 8,121,622 0.13% 2.85 0.40% 91 Azerbaijan 7,911,974 0.12% 2.81 0.39% 92 Switzerland 7,489,370 0.12% 2.74 0.38% 93 Benin 7,460,025 0.12% 2.73 0.38% 94 Bulgaria 7,450,349 0.12% 2.73 0.38% 95 Tajikistan 7,163,506 0.11% 2.68 0.37% 96 Honduras 6,975,204 0.11% 2.64 0.37% 97 Israel 6,955,000 0.11% 2.64 0.37% 98 El Salvador 6,704,932 0.10% 2.59 0.36% 99 Burundi 6,370,609 0.10% 2.52 0.35% 100 Paraguay 6,347,884 0.10% 2.52 0.35% 101 Laos 6,217,141 0.10% 2.49 0.35% 102 Sierra Leone 6,017,643 0.09% 2.45 0.34% 103 Libya 5,765,563 0.09% 2.40 0.33% 104 Jordan 5,759,732 0.09% 2.40 0.33% 105 Togo 5,681,519 0.09% 2.38 0.33% 106 Papua New Guinea 5,545,268 0.09% 2.35 0.33% 107 Nicaragua 5,465,100 0.08% 2.34 0.32% 108 Denmark 5,432,335 0.08% 2.33 0.32% 109 Slovakia 5,431,363 0.08% 2.33 0.32% 110 Finland 5,223,442 0.08% 2.29 0.32% 111 Kyrgyzstan 5,146,281 0.08% 2.27 0.31% 112 Turkmenistan 4,952,081 0.08% 2.23 0.31% 113 Georgia 4,677,401 0.07% 2.16 0.30% 114 Norway 4,593,041 0.07% 2.14 0.30% 115 Eritrea 4,561,599 0.07% 2.14 0.30% 116 Croatia 4,495,904 0.07% 2.12 0.29% 117 Moldova 4,455,421 0.07% 2.11 0.29% 118 Singapore 4,425,720 0.07% 2.10 0.29% 119 Ireland 4,130,700 0.06% 2.03 0.28% 120 New Zealand 4,098,200 0.06% 2.02 0.28% 121 Bosnia and Herzegovina 4,025,476 0.06% 2.01 0.28% 122 Costa Rica 4,016,173 0.06% 2.00 0.28% 123 Lebanon 3,826,018 0.06% 1.96 0.27% 124 Central African Republic 3,799,897 0.06% 1.95 0.27% 125 Lithuania 3,596,617 0.06% 1.90 0.26% 126 Albania 3,563,112 0.06% 1.89 0.26% 127 Liberia 3,482,211 0.05% 1.87 0.26% 128 Uruguay 3,415,920 0.05% 1.85 0.26% 129 Mauritania 3,086,859 0.05% 1.76 0.24% 130 Panama 3,039,150 0.05% 1.74 0.24% 131 Republic of the Congo 3,039,126 0.05% 1.74 0.24% 132 Oman 3,001,583 0.05% 1.73 0.24% 133 Armenia 2,982,904 0.05% 1.73 0.24% 134 Mongolia 2,791,272 0.04% 1.67 0.23% 135 Jamaica 2,731,832 0.04% 1.65 0.23% 136 United Arab Emirates 2,563,212 0.04% 1.60 0.22% 137 Kuwait 2,335,648 0.04% 1.53 0.21% 138 Latvia 2,290,237 0.04% 1.51 0.21% 139 Bhutan 2,232,291 0.03% 1.49 0.21% 140 Macedonia 2,045,262 0.03% 1.43 0.20% 141 Namibia 2,030,692 0.03% 1.43 0.20% 142 Slovenia 2,011,070 0.03% 1.42 0.20% 143 Lesotho 1,867,035 0.03% 1.37 0.19% 144 Botswana 1,640,115 0.03% 1.28 0.18% 145 The Gambia 1,593,256 0.02% 1.26 0.17% 146 Guinea-Bissau 1,416,027 0.02% 1.19 0.16% 147 Gabon 1,389,201 0.02% 1.18 0.16% 148 Estonia 1,332,893 0.02% 1.15 0.16% 149 Mauritius 1,230,602 0.02% 1.11 0.15% 150 Swaziland 1,173,900 0.02% 1.08 0.15% 151 Trinidad and Tobago 1,088,644 0.02% 1.04 0.14% 152 East Timor 1,040,880 0.02% 1.02 0.14% 153 Fiji 893,354 0.01% 0.95 0.13% 154 Qatar 863,051 0.01% 0.93 0.13% 155 Cyprus 780,133 0.01% 0.88 0.12% 156 Guyana 765,283 0.01% 0.87 0.12% 157 Bahrain 688,345 0.01% 0.83 0.12% 158 Comoros 671,247 0.01% 0.82 0.11% 159 Solomon Islands 538,032 0.01% 0.73 0.10% 160 Equatorial Guinea 535,881 0.01% 0.73 0.10% 161 Djibouti 476,703 0.01% 0.69 0.10% 162 Luxembourg 468,571 0.01% 0.68 0.09% 163 Suriname 438,144 0.01% 0.66 0.09% 164 Cape Verde 418,224 0.01% 0.65 0.09% 165 Malta 398,534 0.01% 0.63 0.09% 166 Brunei 372,361 0.01% 0.61 0.08% 167 Maldives 349,106 0.01% 0.59 0.08% 168 The Bahamas 301,790 0.005% 0.55 0.08% 169 Iceland 296,737 0.005% 0.54 0.08% 170 Belize 279,457 0.004% 0.53 0.07% 171 Barbados 279,254 0.004% 0.53 0.07% 172 Vanuatu 205,754 0.003% 0.45 0.06% 173 São Tomé and Príncipe 187,410 0.003% 0.43 0.06% 174 Samoa 177,287 0.003% 0.42 0.06% 175 Saint Lucia 166,312 0.003% 0.41 0.06% 176 Saint Vincent and the Grenadines 117,534 0.002% 0.34 0.05% 177 Tonga 112,422 0.002% 0.34 0.05% 178 Federated States of Micronesia 108,105 0.002% 0.33 0.05% 179 Kiribati 103,092 0.002% 0.32 0.04% 180 Grenada 89,502 0.001% 0.30 0.04% 181 Seychelles 81,188 0.001% 0.28 0.04% 182 Andorra 70,549 0.001% 0.27 0.04% 183 Dominica 69,029 0.001% 0.26 0.04% 184 Antigua and Barbuda 68,722 0.001% 0.26 0.04% 185 Marshall Islands 59,071 0.001% 0.24 0.03% 186 Saint Kitts and Nevis 38,958 0.001% 0.20 0.03% 187 Liechtenstein 33,717 0.001% 0.18 0.03% 188 Monaco 32,409 0.001% 0.18 0.02% 189 San Marino 28,880 0.0004% 0.17 0.02% 190 Palau 20,303 0.0003% 0.14 0.02% 191 Nauru 13,048 0.0002% 0.11 0.02% 192 Tuvalu 11,636 0.0002% 0.11 0.01% 193 Vatican City 921 0.00001% 0.03 0.004%

## Criticisms

It has been claimed that the Penrose method is restricted to votes for which public opinion is equally divided for and against.[7][8][9] 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).[8]

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.[8]

In addition, a minor technical issue 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.[9]