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Abstract. A simple game is a structure G = (N, W) where N= {!,'... ,ri) and W is an arbitrary collection of subsets of A'. 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 e R so that X € W iff ¿2{w(x): x e 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 22 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" or its algebraic analogue known as the Theorem of the Alternative.

Symmetric (3, 2) simple games serve as models for anonymous voting systems in which each voter may vote "yes," abstain, or vote "no," the outcome is "yes" or "no," and all voters play interchangeable roles. The extension to symmetric (j,2) simple games, in which each voter chooses from among j ordered levels of approval, also models some natural decision rules, such as pass-fail grading systems. Each such game is determined by the set of (anonymous) minimal winning profiles. This makes it possible to count the possible systems, and the counts suggest some interesting patterns. In the (3, 2) case, the approach yields a version of May's Theorem, classifying all possible anonymous voting rules with abstention in terms of quota functions. In contrast to the situation for ordinary simple games these results reveal that the class of simple games with 3 or more levels of approval remains large and varied, even after the imposition of symmetry.