A characterization of weighted voting

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∈W iff Σ{w(x) : x∈X}≥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 a , a=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 R n or its algebraic analogue known as the Theorem of the Alternative