Polytopality of Simple Games

The Bier sphere Bier(G)=Bier(K):=K∗ΔK° and the canonical fan Fan(Γ)=Fan(K) are combinatorial/geometric companions of a simple game G=(P,Γ) (equivalently the associated simplicial complex K), where P is the set of players, Γ⊆2P is the set of wining coalitions, and K:=2P∖Γ is the simplicial complex of losing coalitions. We propose and study a general "Steinitz problem" for simple games as the problem of characterizing which games G are polytopal (canonically polytopal) in the sense that the corresponding Bier sphere Bier(G) (fan Fan(Γ)) can be realized as the boundary sphere (normal fan) of a convex polytope. We characterize (roughly) weighted majority games as the games Γ which are canonically (pseudo) polytopal (Theorems 1.1 and 1.2) and show, by an experimental/theoretical argument (Theorem 1.4), that simple games such that Bier(G) is nonpolytopal do not exist in dimension 3. This should be compared to the fact that asymptotically almost all simple games are nonpolytopal and a challenging open problem is to find a nonpolytopal simple game with the smallest number of players.