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2014 | OriginalPaper | Chapter

Square Root Voting System, Optimal Threshold and π

Authors : Karol Życzkowski, Wojciech Słomczyński

Published in: Voting Power and Procedures

Publisher: Springer International Publishing

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Abstract

The problem of designing an optimal weighted voting system for the two-tier voting, applicable in the case of the Council of Ministers of the European Union (EU), is investigated. Various arguments in favor of the square root voting system, where the voting weights of member states are proportional to the square root of their population are discussed and a link between this solution and the random walk in the one-dimensional lattice is established. It is known that the voting power of every member state is approximately equal to its voting weight, if the threshold q for the qualified majority in the voting body is optimally chosen. We analyze the square root voting system for a generic “union” of M states and derive in this case an explicit approximate formula for the level of the optimal threshold: \(q \simeq 1/2 + 1/\sqrt{\pi M}\). The prefactor \(1/\sqrt{\pi }\) appears here as a result of averaging over the ensemble of “unions” with random populations.

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previous chapter A Probabilistic Re-View on Felsenthal and Machover’s “The Measurement of Voting Power”

next chapter The Fate of the Square Root Law for Correlated Voting

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Title: Square Root Voting System, Optimal Threshold and π
Authors: Karol Życzkowski
Wojciech Słomczyński
Publisher: Springer International Publishing
Book: Voting Power and Procedures
Print ISBN: 978-3-319-05157-4

Electronic ISBN: 978-3-319-05158-1

Copyright Year: 2014
DOI: https://doi.org/10.1007/978-3-319-05158-1_8