Polytopality of Simple Games

Abstract

The Bier sphere Bier ( G ) = Bier ( K ) : = K ∗ Δ K ° 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 ⊆ 2 P is the set of wining coalitions, and K : = 2 P ∖ 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.