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.