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Method of allocating voting weight by population
The Penrose method (or square root method) is a method devised in 1946 by Professor Lionel Penrose for allocating the voting weights of delegations (possibly a single representative) in decision making bodies proportional to the square root of the population represented by this delegation. This is justified by the fact that, due to the 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, and for voting in the Council of the European Union.

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". 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). 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. 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.