Penrose's law of square roots

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The square root law of Penrose is a mathematical method for the distribution of seats or votes , which can be applied for example to bodies, where several countries are involved, each uniformly agree or votes (as a block) a proposal to reject him.

In order for every citizen to have the same voting power (power) according to the Banzhaf power index , regardless of the country they come from, the power indices of the countries within the body must be proportional to the square root of the population (1st square root law). The theory was developed in 1946 by the British mathematician Lionel Penrose .

In order to obtain such a distribution of power within the body, the weight of the votes in this method can be chosen proportionally to the square root of the size of the population (2nd law of square roots). However, the distribution is only carried over to the power indices if a suitable approval quorum applies to the votes . An approximation formula for the quorum is

where N i denotes the population sizes. For the EU with 27 states, mathematicians put it at 61.4%. A lower quorum leads to larger, a higher quorum to smaller power indices of the larger states.

The non-governmental organization International Network for a United Nations Second Assembly (INFUSA) has assessed the square root method as

“More than a pragmatic compromise between the two extremes of the distribution of seats without taking into account the size of the population and that of the distribution of seats in direct dependence on the size of the population; Penrose showed that his method according to the laws of statistics would give every voter the same influence on the decisions of a Parliamentary Assembly at the United Nations. "

The method was proposed for the distribution of seats in a reformed EU Council of Ministers and in a Parliamentary Assembly at the United Nations . In the negotiations on the Lisbon Treaty in 2007, the Polish government vehemently (but ultimately unsuccessfully) demanded this type of power distribution for the Council of Ministers.

The American statistician and political scientist Andrew Gelman rejects a distribution of votes based on the square root law. From a statistical analysis of a high number of real elections, he concludes that the assumptions on the behavior of voters on which the square root law is based are not fulfilled in reality and that a fair distribution of the number of votes is based on a high 0.9 law instead of the high 0.5 (square root) would result, so with an almost directly proportional ratio.


  • Lionel Penrose : The elementary statistics of majority voting . In: Journal of the Royal Statistical Society . tape 109 , 1946, pp. 53-57 (English).
  • Dan S. Felsenthal, Moshé Machover: The measurement of voting power. Theory and practice, problems and paradoxes . Elgar, Cheltenham 1998, ISBN 1-85898-805-5 (English).
  • Dan S. Felsenthal, Moshé Machover: Enlargement of the EU and weighted voting in its council of ministers . Project Report. London School of Economics and Political Science, London 2002 (English, ).
  • Wojciech Słomczyński, Karol Życzkowski: Penrose voting system and optimal quota . In: Acta Physica Polonica B . tape 37 , no. 11 , 2006, p. 3133–3143 (English, [PDF; 582 kB ]).

Web links

  • Martin Fehndrich: Square root method,, June 16, 2007/11. October 2007

Individual evidence

  1. Appeal to the United Nations General Assembly to consider the proposal for a UN Second Assembly ( Memento of the original from August 18, 2005 in the Internet Archive ) Info: The @1@ 2Template: Webachiv / IABot / archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. . Earth Rights Institute (English).
  2. ^ EU: Equal Influence for All Citizens , Press Release 151 of the Ruhr University Bochum , May 10, 2004/28. June 2007
  3. Werner Kirsch : Europe, recalculated , Die Zeit , June 9, 2004.
  4. Stefan Tomik: Poland's proposal for square root treatment for Europe . FAZ , June 14, 2007, archived from the original on December 3, 2013 . ;.
  5. ^ Andrew Gelman: Why the square-root rule for vote allocation is a bad idea , October 9, 2007 (English).