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A Characterization of Weighted Voting

Alan Taylor and William Zwicker
Proceedings of the American Mathematical Society
Vol. 115, No. 4 (Aug., 1992), pp. 1089-1094
DOI: 10.2307/2159360
Stable URL: http://www.jstor.org/stable/2159360
Page Count: 6
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
A Characterization of Weighted Voting
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Abstract

A simple game is a structure G = (N, W) where N = {1, ..., n} and W is an arbitrary collection of subsets of N. Sets in W are called winning coalitions and sets not in W are called losing coalitions. G is said to be a weighted voting system if there is a function w: N → R and a "quota" q ∈ R so that $X \in W \iff \sum \{w(x): x \in X\} \geq q$. Weighted voting systems are the hypergraph analogue of threshold graphs. We show here that a simple game is a weighted voting system iff it never turns out that a series of trades among (fewer than 22n not necessarily distinct) winning coalitions can simultaneously render all of them losing. The proof is a self-contained combinatorial argument that makes no appeal to the separating of convex sets in Rn or its algebraic analogue known as the Theorem of the Alternative.

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