Penrose method Totally Explained

Everything about The Penrose Method totally explained

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. It 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's simple to calculate the threshold percentage which constitutes an appropriate "majority", for example what is referred to as the "Jagiellonian Compromise" .
Also many bodies do follow a similar method even though they don't refer to it for example Bundesrat of Germany votes allocation can be approximated as 2.01+Square_root (1.24 * land's population in millions). However this agreement with the Penrose method is accidental, therefore 4 (5 without constraints) out of 16 countries have one vote less than they ought to have according to the Penrose method. But it's an example where political processes converge to game theoretical results.

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'd 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	Republic of 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%

Further Information

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