Polytopality of simple games

• Main Authors: Timotijević, Marinko, Živaljević, Rade T, Jevtić, Filip D
• Format: Journal Article
• Language: English
• Published: 26-09-2023
• Subjects: Mathematics - Combinatorics
• DOI: 10.48550/arxiv.2309.14848

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.