L.S. Penrose's Limit Theorem: Proof of Some Special Cases

Posted: 14 Jun 2005

See all articles by Ines Lindner

Ines Lindner

Free University Amsterdam

Moshé Machover

King’s College London

Abstract

L.S. Penrose was the first to propose a measure of voting power (which later came to be known as 'the (absolute) Banzhaf (Bz) index'). His limit theorem - which is implicit in his booklet (1952) and for which he gave no rigorous proof - says that in simple weighted voting games (WVGs), if the number of voters increases indefinitely while the quota is pegged at half the total weight, then - under certain conditions - the ratio between the voting powers (as measured by him) of any two voters converges to the ratio between their weights. We conjecture that the theorem holds, under rather general conditions, for large classes of variously defined WVGs, other values of the quota, and other measures of voting power. We provide proofs for some special cases.

Keywords: Limit theorems, Majority games, Ternary weighted games, Weighted voting games

JEL Classification: C71, D71

Suggested Citation

Lindner, Ines and Machover, Moshé, L.S. Penrose's Limit Theorem: Proof of Some Special Cases. Available at SSRN: https://ssrn.com/abstract=741567

Ines Lindner (Contact Author)

Free University Amsterdam ( email )

De Boelelaan 1105
1081 HV Amsterdam
Netherlands

Moshé Machover

King’s College London ( email )

Strand
London, England WC2R 2LS
United Kingdom

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