Cube root law

This article is about a trend observed in national legislature sizes relative to the country's population. For the scientific law, see Square–cube law.
A visual representation of the cube root law.

The cube root law is an observation in political science that the number of members of a unicameral legislature, or of the lower house of a bicameral legislature, is about the cube root of the population being represented.[1] The rule was devised by Estonian political scientist Rein Taagepera in his 1972 paper "The size of national assemblies".[2]

The cube root law has been cited in proposals for legislative resizing in the United States and in the 2020 Italian constitutional referendum.

Large deviations between the data and the cube root curve, the inclusion of authoritarian governments in the data, as well as problems with how the cube root curve is derived from underlying postulates, have led researchers to question the cube root law.[3][4]

Derivation

Taagepera's theoretical model says that as the number of legislators expands, the number of constituents per legislator dwindles, and the two numbers must be reconciled.[5] He assumed that for maximum effectiveness the two numbers should be equal.[3] From this theoretical model he derived the equation with as the number of legislators and as the politically active fraction of the total polulation . Assuming that half of the population is politically active led to the simple equation with no coefficient:[2]

Subsequent analysis

Giorgio Margaritondo argued in 2021 that fitting the cube-root equation to the experimental dataset originally used by Taagepera in 1972 leads to the equation:[3] This leads to the problematic result that the politically active population is 14%—too low.[3] He said that although the cube root rule included governments that were corrupt, ineffective, or authoritarian, the rule has been misused as an argument for "optimal" legislature size.[3] Actually, even Taagepera himself said, "there does not seem to be any clear difference in size (at equal population size) between assemblies which are considered representative and those which are considered to be of the rubber-stamp type." He said his model applies to both because "assemblies which are not used for interest aggregation may be used for transmitting government orders to the population."[2] In 2012, researchers Emmanuelle Auriol and Robert J. Gary-Bobo derived a square-root law and empirically supported it from recent data for 100 countries.[4][3] Margaritondo agreed that the data actually fits better to a function with a higher exponent, closer to a square-root law, and that the data's deviation from the cube root rule makes the cube root statistically inferior. In this regard, he gives an optimal formula of:[3] Margaritondo said that actually any power law equation would be so sensitive to the uncertainty in the exponent that the uncertainty in the result is large enough to accommodate most preferences.[3]

Applying this formula to the U.S. House of Representatives as of the 2020 Census would give a House of between 379 and 1231 members, while using an exponent of 0.4507 gives 693 (the same result using the cube root rule).

In 2015 researchers Kristof Jacobs and Simon Otjes questioned the cause-effect sequence used to derive the cube root rule.[3][4] Likewise Margaritondo said that Taagepera should have compared the number of constituents for one legislator with the same legislator's connections to other legislators instead of using the total connections between all legislators. Margaritondo said that the correct logic leads not to a cube-root law but a square-root law.[3]

Political proposals and referenda

The law has led to a proposal to increase the size of the United States House of Representatives so that the number of representatives would be the cube root of the US population as calculated in the most recent census.[6] The House of Representatives has had 435 members since the Reapportionment Act of 1929 was passed; if the US followed the cube root rule, there would be 693 members of the House of Representatives based on the population at the 2020 Census. This proposal was endorsed by the New York Times editorial board in 2018.[7]

The cube root law was used to support one side of the 2020 Italian constitutional referendum.[3]

Table comparing OECD nations in 2019 with EIU Democracy Index ranking

Further information: OECD and Democracy Index

Out of the countries listed, Lithuania is the only one to exactly match the cube root rule. Moreover, Denmark, Canada, Ireland and Mexico come close to matching the rule, while the Netherlands and Australia are counterevidence against the rule.

Some of these countries (e.g. Germany) have overhang seats in a mixed member proportional system, as a result the size of their parliaments can vary significantly between elections.

Country Lower or unicameral house Population (2019)[8] Lower house size (2019) Cube root of population (nearest person) Difference between lower house and cube root of population Difference between lower house and cube root of population (%) People per representative People per representative (cube root lower house) Democracy Index Ranking (2022)[9]
Australia House of Representatives 25,364,307 151 294 −143 −49% 167,976 86,327 15
Austria National Council 8,877,067 183 207 −24 −12% 48,509 42,873 20
Belgium Chamber of Representatives 11,484,055 150 226 −76 −34% 76,560 50,901 36
Canada House of Commons 37,589,262 338 335 +3 +1% 111,211 112,213 12
Chile Chamber of Deputies 18,952,038 155 267 −112 −42% 122,271 71,084 19
Colombia Chamber of Representatives 50,339,443 172 363 −191 −53% 303,250 136,334 53
Czech Republic Chamber of Deputies 10,669,709 200 220 −20 −9% 53,349 48,466 25
Denmark Folketing 5,818,553 179 180 −1 −1% 32,506 32,350 6
Estonia Riigikogu 1,326,590 101 110 −9 −8% 13,135 12,073 27
Finland Parliament 5,520,314 200 177 +23 +13% 27,602 31,235 5
France National Assembly 67,059,887 577 406 +171 +42% 116,222 165,060 22
Germany Bundestag 83,132,799 734 436 +298 +68% 113,260 190,480 14
Greece Parliament 10,716,322 300 220 +80 +36% 35,721 48,607 25
Hungary National Assembly 9,769,949 199 214 −15 −7% 49,095 45,701 56
Iceland Althing 361,313 63 71 −8 −11% 5,735 5,073 3
Ireland Dáil 5,123,536 174 172 +2 +1% 29,446 29,788 8
Israel Knesset 9,053,300 120 208 −88 −42% 75,444 43,438 29
Italy Chamber of Deputies 60,297,396 400 392 +8 +2% 150,743 153,768 34
Japan House of Representatives 126,264,931 465 502 −37 −7% 271,537 251,684 16
Korea, Republic of National Assembly 51,709,098 300 373 −73 −20% 172,384 138,796 24
Latvia Saeima 1,912,789 100 124 −24 −19% 19,218 15,409 38
Lithuania Seimas 2,786,844 141 141 0 0% 19,765 19,803 39
Luxembourg Chamber of Deputies 619,896 60 85 −25 −29% 10,332 7,270 13
Mexico Chamber of Deputies 127,575,529 500 503 −3 −1% 255,151 253,422 89
Netherlands House of Representatives 17,332,850 150 259 −109 −42% 115,552 66,975 9
New Zealand House of Representatives 4,917,000 120 170 −50 −29% 40,975 28,916 2
Norway Storting 5,347,896 169 175 −6 −3% 31,644 30,581 1
Poland Sejm 37,970,874 460 336 +124 +37% 82,545 112,971 46
Portugal Assembly of the Republic 10,269,417 230 217 +13 +6% 44,650 47,246 28
Slovakia National Council 5,454,073 150 176 −26 −15% 36,360 30,985 43
Slovenia National Assembly 2,087,946 90 128 −38 −30% 23,199 16,336 31
Spain Congress of Deputies 47,076,781 350 361 −11 −3% 134,505 130,378 22
Sweden Riksdag 10,285,453 349 217 +132 +61% 29,471 47,295 4
Switzerland National Council 8,574,832 200 205 −5 −2% 42,874 41,894 7
Turkey Grand National Assembly 83,429,615 600 437 +163 +37% 139,049 190,933 103
United Kingdom House of Commons 66,834,405 650 406 +244 +60% 102,822 164,690 18
United States House of Representatives 328,239,523 435 690 −255 −37% 754,574 475,840 30

Historical US House sizes

The following table describes how the US House of Representatives would have looked historically under the cube root rule according to the Huntington–Hill method.

Census, Year Size AL AK AZ AR CA CO CT DE DC FL GA HI ID IL IN IA KS KY LA ME MD MA MI MN MS MO MT NE NV NH NJ NM NY NC ND OH OK OR PA RI SC SD TN TX UT VT VA WA WV WI WY
1st, 1790 158 10 3 3 13 20 6 8 14 16 18 3 10 34
2nd, 1800 175 8 2 5 7 12 19 6 7 20 16 20 2 12 4 5 30
3rd, 1810 194 7 2 7 11 11 19 6 7 27 15 6 22 2 12 7 6 27
4th, 1820 213 3 6 2 8 1 3 13 3 7 9 12 2 5 6 31 14 13 24 2 11 9 5 24
5th, 1830 235 6 5 1 10 3 6 13 4 7 8 11 3 3 5 6 35 14 17 25 2 11 13 5 22
6th, 1840 258 9 2 5 1 11 7 10 12 5 8 7 11 3 6 6 4 6 37 12 23 26 2 9 13 4 19
7th, 1850 286 10 3 1 5 1 1 11 10 12 2 12 6 7 7 12 5 8 8 4 6 38 11 25 29 2 8 12 3 4 18 4
8th, 1860 316 10 4 4 5 1 2 11 17 14 7 12 7 6 7 13 8 2 8 12 3 7 39 10 24 1 29 2 7 11 6 3 16 8
9th, 1870 338 9 4 5 5 1 2 10 22 15 11 3 12 6 6 7 13 10 4 7 15 1 1 3 8 39 10 24 1 31 2 6 11 7 3 11 4 9
10th, 1880 369 9 6 6 2 5 1 2 12 23 15 12 7 12 7 5 7 13 12 6 8 16 3 1 3 8 38 11 24 1 32 2 7 12 12 3 11 5 10
11th, 1890 398 10 7 8 3 5 1 3 12 1 25 14 12 9 12 7 4 7 14 13 8 8 17 1 7 1 2 9 39 10 1 24 2 34 2 7 2 11 14 2 11 2 5 11 1
12th, 1900 426 10 7 8 3 5 1 2 3 13 1 27 14 13 8 12 8 4 7 16 14 10 9 18 1 6 1 2 11 41 11 2 24 2 36 2 8 2 11 17 2 2 11 3 5 12 1
13th, 1910 453 11 8 12 4 6 1 2 4 13 2 28 13 11 8 11 8 4 6 17 14 10 9 16 2 6 1 2 12 45 11 3 23 8 3 38 3 7 3 11 19 2 2 10 6 6 12 1
14th, 1920 475 11 2 8 15 4 6 1 2 4 13 2 29 13 11 8 11 8 3 6 17 16 11 8 15 3 6 1 2 14 2 46 11 3 26 9 4 39 3 8 3 10 21 2 2 10 6 7 12 1
15th, 1930 500 11 2 7 23 4 6 1 2 6 12 2 31 13 10 8 11 8 3 7 17 20 10 8 15 2 6 1 2 16 2 51 13 3 27 10 4 39 3 7 3 11 23 2 2 10 6 7 12 1
16th, 1940 512 11 2 8 27 4 7 1 3 7 12 2 31 13 10 7 11 9 3 7 17 20 11 9 15 2 5 1 2 16 2 52 14 3 27 9 4 39 3 7 3 11 25 2 1 10 7 7 12 1
17th, 1950 536 11 3 7 38 5 7 1 3 10 12 2 31 14 9 7 10 10 3 8 17 23 11 8 14 2 5 1 2 17 2 53 14 2 28 8 5 37 3 8 2 12 27 2 1 12 8 7 12 1
18th, 1960 566 10 1 4 6 49 6 8 1 2 16 12 2 2 32 15 9 7 10 10 3 10 16 25 11 7 14 2 4 1 2 19 3 53 14 2 31 7 6 36 3 8 2 11 30 3 1 12 9 6 12 1
19th, 1970 590 10 1 5 6 58 6 9 2 2 20 13 2 2 32 15 8 7 9 11 3 11 17 26 11 6 14 2 4 2 2 21 3 53 15 2 31 7 6 34 3 8 2 11 32 3 1 13 10 5 13 1
20th, 1980 612 11 1 7 6 64 8 8 2 2 26 15 3 3 31 15 8 6 10 11 3 11 16 25 11 7 13 2 4 2 3 20 4 48 16 2 29 8 7 32 3 8 2 12 38 4 1 14 11 5 13 1
21st, 1990 631 10 1 9 6 75 8 8 2 2 33 16 3 3 29 14 7 6 9 11 3 12 15 24 11 7 13 2 4 3 3 20 4 46 17 2 27 8 7 30 3 9 2 12 43 4 2 16 12 5 12 1
22nd, 2000 657 10 2 12 6 79 10 8 2 1 37 19 3 3 29 14 7 6 9 10 3 12 15 23 11 7 13 2 4 5 3 20 4 44 19 2 27 8 8 29 2 9 2 13 49 5 2 17 14 4 13 1
23rd, 2010 677 10 2 14 6 82 11 8 2 1 41 21 3 3 28 14 7 6 10 10 3 13 14 22 12 7 13 2 4 6 3 19 5 43 21 2 25 8 8 28 2 10 2 14 55 6 1 18 15 4 12 1
24th, 2020 695 11 2 15 6 83 12 8 2 2 45 22 3 4 27 14 7 6 9 10 3 13 15 21 12 6 13 2 4 7 3 19 4 42 22 2 25 8 9 27 2 11 2 15 61 7 1 18 16 4 12 1
Census, Year Size AL AK AZ AR CA CO CT DE DC FL GA HI ID IL IN IA KS KY LA ME MD MA MI MN MS MO MT NE NV NH NJ NM NY NC ND OH OK OR PA RI SC SD TN TX UT VT VA WA WV WI WY

See also

References

  1. ^ Lutz, Donald S. (2006). Principles of Constitutional Design. Cambridge University Press. ISBN 9781139460552.
  2. ^ a b c Taagepera, Rein (1972). "The size of national assemblies". Social Science Research. 1 (4): 385–401. doi:10.1016/0049-089X(72)90084-1.
  3. ^ a b c d e f g h i j k Margaritondo, Giorgio (2021). "Size of National Assemblies: The Classic Derivation of the Cube-Root Law is Conceptually Flawed". Frontiers in Physics. 8: 606. Bibcode:2021FrP.....8..606M. doi:10.3389/fphy.2020.614596. ISSN 2296-424X.
  4. ^ a b c Papageorgiou, Nik. "Physics challenges the optimal size of parliaments". EPFL. Retrieved February 10, 2026.
  5. ^ De Sio, Lorenzo. "945 sono troppi? 600 sono pochi? Qual è il numero "ottimale" di parlamentari?". Centro Italiano Studi Elettorali. Retrieved February 12, 2026.
  6. ^ Kane, Caroline; Mascioli, Gianni; McGarry, Michael; Nagel, Meira (January 2020). Why the House of Representatives Must Be Expanded and How Today's Congress Can Make It Happen (PDF) (Report). Fordham University School of Law. Retrieved 17 September 2020.
  7. ^ "America Needs a Bigger House". New York Times. 9 November 2018. Retrieved 17 September 2020.
  8. ^ "Population, total - OECD members | Data". data.worldbank.org. Retrieved 2020-09-19.
  9. ^ "EIU Report: Democracy Index 2022". Economist Intelligence Unit. 2023. Retrieved April 24, 2023.
allikas: https://en.wikipedia.org/wiki/Cube_root_law