In the mathematical theory of games, the Penrose square root law, originally formulated by

Lionel Penrose Lionel Sharples Penrose, FRS (11 June 1898 – 12 May 1972) was an English psychiatrist, medical geneticist, paediatrician, mathematician and chess theorist, who carried out pioneering work on the genetics of intellectual disability. Penrose w ...

, concerns the distribution of the voting power in a voting body consisting of ''N'' members. It states that the ''a priori'' voting power of any voter, measured by the Penrose–Banzhaf index

\psi

scales like

1/\sqrt

. This result was used to design the

Penrose method 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 ...

for allocating the voting weights of representatives in a decision-making bodies proportional to the square root of the population represented.

Short derivation

To estimate the voting index of any player one needs to estimate the number of the possible winning coalitions in which his vote is decisive. Assume for simplicity that the number of voters is odd, ''N'' = 2''j'' + 1, and the body votes according to the standard majority rule. Following Penrose one concludes that a given voter will be able to effectively influence the outcome of the voting only if the votes split half and half: if ''j'' players say 'Yes' and the remaining ''j'' players vote 'No', the last vote is decisive. Assuming that all members of the body vote independently (the votes are uncorrelated) and that the probability of each vote 'Yes' is equal to ''p'' = 1/2 one can estimate likelihood of such an event using the

Bernoulli trial In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is c ...

. The

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

to obtain ''j'' votes 'Yes' out of 2''j'' votes reads :

P_j = \left(  \frac\right)  ^\frac.

For large ''N'' we may use the

Stirling's approximation In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less p ...

for the factorial ''j'' ! and obtain the probability

\psi

that the vote of a given voter is decisive :

\psi=P_j \sim 2^\frac\ =\ \frac \sim \sqrt \frac.

The same approximation is obtained for an even number ''N''. A mathematical investigation of the influence of possible correlations between the voters for the Penrose square root law was presented by Kirsch. Penrose law is applied to construct Penrose-like systems of two-tier voting, including the Jagiellonian Compromise designed for the

Council of the European Union The Council of the European Union, often referred to in the treaties and other official documents simply as the Council, and informally known as the Council of Ministers, is the third of the seven Institutions of the European Union (EU) as ...

References

{{reflist, 30em Game theory

Short derivation

See also

References