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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A characterization of weighted voting


Authors: Alan Taylor and William Zwicker
Journal: Proc. Amer. Math. Soc. 115 (1992), 1089-1094
MSC: Primary 90A28
DOI: https://doi.org/10.1090/S0002-9939-1992-1092927-0
MathSciNet review: 1092927
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Abstract: A simple game is a structure $ G = (N,W)$ where $ N = \{ 1, \ldots ,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 \to \mathbb{R}$ and a "quota" $ q \in \mathbb{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 $ {2^{{2^n}}}$ 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 $ {\mathbb{R}^n}$ or its algebraic analogue known as the Theorem of the Alternative.


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DOI: https://doi.org/10.1090/S0002-9939-1992-1092927-0
Article copyright: © Copyright 1992 American Mathematical Society