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. The rule was devised by Estonian political scientist Rein Taagepera in his 1972 paper "The size of national assemblies".

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.

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. He assumed that for maximum effectiveness the two numbers should be equal. From this theoretical model he derived the equation with as the number of legislators and as the politically active fraction of the total population . Assuming that half of the population is politically active led to the simple equation with no coefficient:

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: This leads to the problematic result that the politically active population is 14%, which is too low. 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. 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." 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. 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: 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.

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. 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.

Political proposals and referendums 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. 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.

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

Table comparing OECD nations in 2019 with EIU Democracy Index ranking 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.

+CountryLower or unicameral housePopulation (2019)Lower house size (2019)Cube root of population (nearest person)Difference between lower house and cube root of populationDifference between lower house and cube root of population (%)People per representativePeople per representative (cube root lower house)Democracy Index Ranking (2022)AustraliaHouse of Representatives25,364,307151294−143−49%167,97686,32715AustriaNational Council8,877,067183207−24−12%48,50942,87320BelgiumChamber of Representatives11,484,055150226−76−34%76,56050,90136CanadaHouse of Commons37,589,262338335+3+1%111,211112,21312ChileChamber of Deputies18,952,038155267−112−42%122,27171,08419ColombiaChamber of Representatives50,339,443172363−191−53%303,250136,33453Czech RepublicChamber of Deputies10,669,709200220−20−9%53,34948,46625DenmarkFolketing5,818,553179180−1−1%32,50632,3506EstoniaRiigikogu1,326,590101110−9−8%13,13512,07327FinlandParliament5,520,314200177+23+13%27,60231,2355FranceNational Assembly67,059,887577406+171+42%116,222165,06022GermanyBundestag83,132,799734436+298+68%113,260190,48014GreeceParliament10,716,322300220+80+36%35,72148,60725HungaryNational Assembly9,769,949199214−15−7%49,09545,70156IcelandAlthing361,3136371−8−11%5,7355,0733IrelandDáil5,123,536174172 +2 +1%29,44629,7888IsraelKnesset9,053,300120208−88−42%75,44443,43829ItalyChamber of Deputies60,297,396400392+8+2%150,743153,76834JapanHouse of Representatives126,264,931465502−37−7%271,537251,68416Korea, Republic ofNational Assembly51,709,098300373−73−20%172,384138,79624LatviaSaeima1,912,789100124−24−19%19,21815,40938LithuaniaSeimas2,786,84414114100%19,76519,80339LuxembourgChamber of Deputies619,8966085−25−29%10,3327,27013MexicoChamber of Deputies127,575,529500503−3−1%255,151253,42289NetherlandsHouse of Representatives17,332,850150259−109−42%115,55266,9759New ZealandHouse of Representatives4,917,000120170−50−29%40,97528,9162NorwayStorting5,347,896169175−6−3%31,64430,5811PolandSejm37,970,874460336+124+37%82,545112,97146PortugalAssembly of the Republic10,269,417230217+13+6%44,65047,24628SlovakiaNational Council5,454,073150176−26−15%36,36030,98543SloveniaNational Assembly2,087,94690128−38−30%23,19916,33631SpainCongress of Deputies47,076,781350361−11−3%134,505130,37822SwedenRiksdag10,285,453349217+132+61%29,47147,2954SwitzerlandNational Council8,574,832200205−5−2%42,87441,8947TurkeyGrand National Assembly83,429,615600437+163+37%139,049190,933103United KingdomHouse of Commons66,834,405650406+244+60%102,822164,69018United StatesHouse of Representatives328,239,523435690−255−37%754,574475,84030

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 SizeALAKAZARCACOCTDEDCFLGAHIIDILINIAKSKYLAMEMDMAMIMNMSMOMTNENVNHNJNMNYNCNDOHOKORPARISCSDTNTXUTVTVAWAWVWIWY 1st, 1790 158103 313206814161831034 2nd, 1800 17582 571219672016202124530 3rd, 1810 19472 71111196727156222127627 4th, 1820 213362 8131337912256311413242119524 5th, 1830 235651 1036134781133563514172521113522 6th, 1840 2589251 11710125871136646371223262913419 7th, 1850 286103151 1111012212677125884638112529281234184 8th, 1860 316104451 2111714712767138281237391024129271163168 9th, 1870 33894551 2102215113126671310471511383910241312611731149 10th, 1880 369966251 212231512712757131268163138381124132271212311510 11th, 1890 3981078351 312125141291274714138817171293910124234272111421125111 12th, 1900 42610783512313127141381284716141091816121141112242362821117221135121 13th, 1910 4531181246124132281311811846171410916261212451132383383731119221066121 14th, 1920 475112815461241322913118118361716118153612142461132694393831021221067121 15th, 1930 5001127234612612231131081183717201081526121625113327104393731123221067121 16th, 1940 512112827471371223113107119371720119152512162521432794393731125211077121 17th, 1950 5361137385713101223114971010381723118142512172531422885373821227211287121 18th, 1960 5661014649681216122232159710103101625117142412193531423176363821130311296121 19th, 1970 5901015658692220132232158791131117261161424222135315231763438211323113105131 20th, 1980 61211176648822261533311586101131116251171324232044816229873238212384114115131 21st, 1990 6311019675882233163329147691131215241171324332044617227873039212434216125121 22nd, 2000 65710 212 67910 8 213719 3 32914 7 6 910 312152311 713 2 4 5 320 44419 227 8 829 2 9 21349 5 21714 413 1 23rd, 2010 67710 214 68211 8 214121 3 32814 7 61010 313142212 713 2 4 6 319 54321 225 8 828 210 21455 6 11815 412 1 24th, 2020 69511 215 68312 8 224522 3 42714 7 6 910 313152112 613 2 4 7 319 44222 225 8 927 211 21561 7 11816 412 1 Census, Year Size AL AK AZ AR CA CO CT DE DCFL 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 List of legislatures by number of members United States congressional apportionment Wyoming Rule Apportionment in the European Parliament

References