View source for Social:Penrose method

{{short description|Method of allocating voting weight by population}}
The '''Penrose method''' (or '''square-root method''') is a method devised in 1946 by Professor Lionel Penrose<ref name=PenroseOriginalPaper>{{cite journal | author=L.S. Penrose | title=The elementary statistics of majority voting | journal=Journal of the Royal Statistical Society | volume=109 | pages=53–57 | year=1946 | issue=1 | url=http://www.ww.uni-magdeburg.de/bizecon/material/Penrose_The%20Elementary%20Statistics%20of%20Majority%20Voting_Journal%20of%20the%20Royal%20Statistical%20Society_1091)1946_53-57.pdf | doi=10.2307/2981392| jstor=2981392 }}</ref> for allocating the voting weights of delegations (possibly a single representative) in decision-making bodies proportional to the [[Square root|square root]] of the population represented by this delegation. This is justified by the fact that, due to the [[Penrose square root law|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,<ref name=PenroseOriginalPaper/><ref name=INFUSA>{{cite web | title = Proposal for a United Nations Second Assembly | publisher = International Network for a UN Second Assembly | year = 1987 | url = http://www.earthrights.net/archives/gpa/unsa.html | accessdate = 27 April 2010}}</ref> and for voting in the Council of the European Union.<ref name="Penrose Voting System and Optimal Quota">{{cite journal|author=W. Slomczynski, K. Zyczkowski|title=Penrose Voting System and Optimal Quota|journal=Acta Physica Polonica B|volume=37|pages=3133–3143|year=2006|issue=11|arxiv=physics/0610271 |bibcode=2006AcPPB..37.3133S |url=http://chaos.if.uj.edu.pl/~karol/pdf/SZapp06.pdf }}</ref><ref name=xyz>{{cite web | title =Maths tweak required for EU voting | publisher =BBC News | date =7 July 2004 | url =http://news.bbc.co.uk/2/hi/science/nature/3804841.stm | accessdate = 27 April 2011}}</ref>

==The EU proposal==

{| class="wikitable" style="float:right; font-size:80%; text-align:center; valign:bottom; margin-left:1em; margin-bottom:1em;"
|+ <big>Comparison of voting weights</big><br/> ''Population in millions as of 1 January 2003 <ref>{{Cite journal | url=http://www.eds-destatis.de/en/downloads/sif/nk_04_13.pdf | title=First results of the demographic data collection for 2003 in Europe | journal=Statistics in Focus: Population and Social Conditions: 13/2004 | publisher=Joint demographic data collection the Council of Europe and Eurostat | author=François-Carlos Bovagnet | year=2004 | accessdate= 28 April 2011}}</ref>''
|- style="background: #ffebad;"
! Member state
| colspan="2" | Population
| colspan="2" | Nice
| colspan="2" | Penrose<ref name="Penrose Voting System and Optimal Quota"/>
|-
| style="text-align:left;" | {{Flag|Germany}}
| style="background: #efefef;" | 82.54m
| 16.5%
| style="background: #efefef;" | 29
| 8.4%
| style="background: #efefef;" | 9.55%
|-
| style="text-align:left;" | {{Flag|France}}
| style="background: #efefef;" | 59.64m
| 12.9%
| style="background: #efefef;" | 29
| 8.4%
| style="background: #efefef;" | 8.11%
|-
| style="text-align:left;" | {{Flag|UK}}
| style="background: #efefef;" | 59.33m
| 12.4%
| style="background: #efefef;" | 29
| 8.4%
| style="background: #efefef;" | 8.09%
|-
| style="text-align:left;" | {{Flag|Italy}}
| style="background: #efefef;" | 57.32m
| 12.0%
| style="background: #efefef;" | 29
| 8.4%
| style="background: #efefef;" | 7.95%
|-
| style="text-align:left;" | {{Flag|Spain}}
| style="background: #efefef;" | 41.55m
| 9.0%
| style="background: #efefef;" | 27
| 7.8%
| style="background: #efefef;" | 6.78%
|-
| style="text-align:left;" | {{Flag|Poland}}
| style="background: #efefef;" | 38.22m
| 7.6%
| style="background: #efefef;" | 27
| 7.8%
| style="background: #efefef;" | 6.49%
|-
| style="text-align:left;" | {{Flag|Romania}}
| style="background: #efefef;" | 21.77m
| 4.3%
| style="background: #efefef;" | 14
| 4.1%
| style="background: #efefef;" | 4.91%
|-
| style="text-align:left;" | {{Flag|Netherlands}}
| style="background: #efefef;" | 16.19m
| 3.3%
| style="background: #efefef;" | 13
| 3.8%
| style="background: #efefef;" | 4.22%
|-
| style="text-align:left;" | {{Flag|Greece}}
| style="background: #efefef;" | 11.01m
| 2.2%
| style="background: #efefef;" | 12
| 3.5%
| style="background: #efefef;" | 3.49%
|-
| style="text-align:left;" | {{Flag|Portugal}}
| style="background: #efefef;" | 10.41m
| 2.1%
| style="background: #efefef;" | 12
| 3.5%
| style="background: #efefef;" | 3.39%
|-
| style="text-align:left;" | {{Flag|Belgium}}
| style="background: #efefef;" | 10.36m
| 2.1%
| style="background: #efefef;" | 12
| 3.5%
| style="background: #efefef;" | 3.38%
|-
| style="text-align:left;" | {{Flag|Czech Republic|name=Czech Rep.}}
| style="background: #efefef;" | 10.20m
| 2.1%
| style="background: #efefef;" | 12
| 3.5%
| style="background: #efefef;" | 3.35%
|-
| style="text-align:left;" | {{Flag|Hungary}}
| style="background: #efefef;" | 10.14m
| 2.0%
| style="background: #efefef;" | 12
| 3.5%
| style="background: #efefef;" | 3.34%
|-
| style="text-align:left;" | {{Flag|Sweden}}
| style="background: #efefef;" | 8.94m
| 1.9%
| style="background: #efefef;" | 10
| 2.9%
| style="background: #efefef;" | 3.14%
|-
| style="text-align:left;" | {{Flag|Austria}}
| style="background: #efefef;" | 8.08m
| 1.7%
| style="background: #efefef;" | 10
| 2.9%
| style="background: #efefef;" | 2.98%
|-
| style="text-align:left;" | {{Flag|Bulgaria}}
| style="background: #efefef;" | 7.85m
| 1.5%
| style="background: #efefef;" | 10
| 2.9%
| style="background: #efefef;" | 2.94%
|-
| style="text-align:left;" | {{Flag|Denmark}}
| style="background: #efefef;" | 5.38m
| 1.1%
| style="background: #efefef;" | 7
| 2.0%
| style="background: #efefef;" | 2.44%
|-
| style="text-align:left;" | {{Flag|Slovakia}}
| style="background: #efefef;" | 5.38m
| 1.1%
| style="background: #efefef;" | 7
| 2.0%
| style="background: #efefef;" | 2.44%
|-
| style="text-align:left;" | {{Flag|Finland}}
| style="background: #efefef;" | 5.21m
| 1.1%
| style="background: #efefef;" | 7
| 2.0%
| style="background: #efefef;" | 2.39%
|-
| style="text-align:left;" | {{Flag|Ireland}}
| style="background: #efefef;" | 3.96m
| 0.9%
| style="background: #efefef;" | 7
| 2.0%
| style="background: #efefef;" | 2.09%
|-
| style="text-align:left;" | {{Flag|Lithuania}}
| style="background: #efefef;" | 3.46m
| 0.7%
| style="background: #efefef;" | 7
| 2.0%
| style="background: #efefef;" | 1.95%
|-
| style="text-align:left;" | {{Flag|Latvia}}
| style="background: #efefef;" | 2.33m
| 0.5%
| style="background: #efefef;" | 4
| 1.2%
| style="background: #efefef;" | 1.61%
|-
| style="text-align:left;" | {{Flag|Slovenia}}
| style="background: #efefef;" | 2.00m
| 0.4%
| style="background: #efefef;" | 4
| 1.2%
| style="background: #efefef;" | 1.48%
|-
| style="text-align:left;" | {{Flag|Estonia}}
| style="background: #efefef;" | 1.36m
| 0.3%
| style="background: #efefef;" | 4
| 1.2%
| style="background: #efefef;" | 1.23%
|-
| style="text-align:left;" | {{Flag|Cyprus}}
| style="background: #efefef;" | 0.72m
| 0.2%
| style="background: #efefef;" | 4
| 1.2%
| style="background: #efefef;" | 0.89%
|-
| style="text-align:left;" | {{Flag|Luxembourg}}
| style="background: #efefef;" | 0.45m
| 0.1%
| style="background: #efefef;" | 4
| 1.2%
| style="background: #efefef;" | 0.70%
|-
| style="text-align:left;" | {{Flag|Malta}}
| style="background: #efefef;" | 0.40m
| 0.1%
| style="background: #efefef;" | 3
| 0.9%
| style="background: #efefef;" | 0.66%
|- style="background:#cccccc;"
| style="text-align:left;" | {{Flag|EU}}
| 484.20m
| 100%
| 345
| 100%
| 100%
|}

The Penrose method became revitalised within the [[Social:European Union|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%.<ref name="Penrose Voting System and Optimal Quota"/> After the university of the authors of this paper, this system is referred to as the "[[Social:Jagiellonian Compromise|Jagiellonian Compromise]]". Optimal threshold decreases with the number <math> M </math> 
of the member states as   <math> 1/2 +1/\sqrt{\pi M} </math>.<ref name="srvs">{{cite book|author=K. Zyczkowski, W. Slomczynski|title=Power, Voting, and Voting Power: 30 Years After |chapter=Square Root Voting System, Optimal Threshold and $$ \uppi $$ π |year=2013 |pages=573–592 |doi=10.1007/978-3-642-35929-3_30 |arxiv=1104.5213|isbn=978-3-642-35928-6 |s2cid=118756505 }}</ref>

==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".<ref name=INFUSA/>

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-[[Social:World War II|World War II]] population figures.<ref name=PenroseOriginalPaper/>

<div align=center>
{| class="wikitable"
|- bgcolor="#ececec" valign=top
| colspan=2 |
| '''Population<br>as of 2005'''
| '''Percent of<br>world population'''
| '''Voting weight'''
| '''Percent of<br>total weight'''
|- 
| 
| '''World'''
| '''6,434,577,575'''
| '''100.00%'''
| '''721.32'''
| '''100.00%'''
|- bgcolor="#ececec" valign=top
| '''Rank'''
| '''Country'''
| colspan=4 |
|- 
| 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%
|- 
|}</div>

==Criticisms==
It has been claimed that the [[Penrose square root law]] is limited to votes for which public opinion is equally divided for and against.<ref name=GelmanWeb>{{cite web | last=Gelman | first =Andrew |  title =Why the square-root rule for vote allocation is a bad idea | work =Statistical Modeling, Causal Inference, and Social Science | publisher =[[Organization:Columbia University|Columbia University]] website | date =9 October 2007 | url =http://www.stat.columbia.edu/~cook/movabletype/archives/2007/10/why_the_squarer.html | accessdate =30 April 2011 }}</ref><ref name="Gelman">{{cite journal|author=Gelman, Katz and Bafumi|title=Standard Voting Power Indexes Do Not Work: An Empirical Analysis|journal=British Journal of Political Science|volume=34|pages=657–674|year=2004|issue=4|url=http://www.stat.columbia.edu/~gelman/research/published/gelmankatzbafumi.pdf|doi=10.1017/s0007123404000237|s2cid=14287710}}</ref><ref name="on the j comp">[http://www.inference.phy.cam.ac.uk/ear23/voting/voting.pdf On the "Jagiellonian compromise"]</ref> 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).<ref name="Gelman"/>

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, which suggests that no single vote is pivotal.<ref name="Gelman"/>

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.<ref name="on the j comp"/>

==See also==
* List of countries by population

== References ==
<references/>

==External links==
*[http://ideas.repec.org/p/wrk/warwec/824.html 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 [https://www.ams.org/featurecolumn/archive/weighted6.html here].

[[Category:Electoral systems]]

{{Sourceattribution|Penrose method}}