Polytopality of simple games

TL;DR

This paper explores the geometric and combinatorial structures of simple games, characterizing roughly weighted majority games through polytopality of associated complexes, and demonstrates that all simple games with up to five players are polytopal.

Contribution

It provides a characterization of roughly weighted majority games via polytopality of Bier spheres and fans, and shows all small simple games are polytopal.

Findings

All simple games with up to five players are polytopal.

Roughly weighted majority games correspond to polytopal Bier spheres and fans.

The study combines experimental and theoretical methods to analyze polytopality.

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.