A characterization of weighted voting

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.