Polytopality of simple games

Abstract: The Bier sphere $Bier(\mathcal{G}) = Bier(K) = K\ast_\Delta K^\circ$ and the canonical fan $Fan(\Gamma) = Fan(K)$ are combinatorial/geometric companions of a simple game $\mathcal{G} = (P,\Gamma)$ (equivalently the associated simplicial complex $K$), where $P$ is the set of players, $\Gamma\subseteq 2^P$ is the set of wining coalitions, and $K = 2^P\setminus \Gamma$ is the simplicial complex of losing coalitions. We characterize roughly weighted majority games as the games $\Gamma$ such that $Bier(\mathcal{G})$ (respectively $Fan(\Gamma)$) is canonically polytopal (canonically pseudo-polytopal) and show, by an experimental/theoretical argument, that all simple games with at most five players are polytopal.