Polytopality of simple games

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.