Penrose method: Wikis

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The Penrose method is a method, devised in 1946 by Professor Lionel Penrose, for allocating seats or votes in legislatures based on the square root of the population of the representative's district or state.

This allocation is motivated by Penrose's work suggesting that the voting power of a representative (defined by the Penrose-Banzhaf index) is inversely proportional to the square root of the number of voters he represents. This theory has become known as the Penrose square root law. Under this theory, if the voting power of each representative scales like square root of the population he represents, the voting power of all voters in all districts is equal.

The voting system of Penrose has been proposed as a method for apportioning representation in a United Nations Parliamentary Assembly, and for voting in the Council of the European Union.

Recently, the Penrose method became revitalised, as it was proposed by Sweden in 2003 (negotiations on the Amsterdam Treaty) and Poland in 2007 (the June 2007 summit on the Treaty of Lisbon) as method of computing voting power of each state in the European Union. With modern computers, it is simple to calculate the threshold percentage which constitutes an optimal level of qualified "majority", at which the voting powers of all citizens in any member state are equal. The level of the optimal threshold, (about 61.6% for EU-27) decreases with the number of the member states. This system is referred to as the "Jagiellonian Compromise" [1 ] [2].

Criticisms

It has been claimed that the Penrose method is restricted only to votes for which public opinion is equally divided for and against[3][4]. 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)[3].

In addition, 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[4].

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 14 most populous nations get fewer seats than they would under one man, one vote; the other nations get more seats.

 Population as of 2005-12-15 Percent of world population Seats Percent of total seats 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 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%

References

1. ^ W. Slomczynski, K. Zyczkowski (2006). "Penrose Voting System and Optimal Quota". Acta Physica Polonica B 37 (11): 3133.
2. ^ Physics World 2006; 19(3):35-37.
3. ^ a b Gelman, Katz and Bafumi (2004). "Standard Voting Power Indexes Do Not Work: An Empirical Analysis". British Journal of Political Science 34: 657-674.
4. ^ a b On the "Jagiellonian compromise"

• Penrose, L., The elementary statistics of majority voting, J. of the Royal Statistical Society, 109 (1946) 53-57.
• article by W. Kirsch at Center for European Policy Studies
• article by D. Leech and H. Aziz at University of Warwick
• many more references at the web page of American Mathematical Society here.