Weighted simple games and the topology of simplicial complexes

TL;DR

This paper employs topological methods using simplicial complexes to analyze simple and weighted voting games, providing new characterizations and formulas for power indices, with practical examples like the U.S. Electoral College.

Contribution

It introduces a topological framework for modeling and analyzing weighted voting games with impossible coalitions, including explicit formulas for power indices.

Findings

Topological characterizations of simple and weighted games

Formulas for Banzhaf and Shapley-Shubik indices in weighted games

Application to real-world voting systems like the Electoral College

Abstract

We use simplicial complexes to model simple games as well as weighted voting games where certain coalitions are considered impossible. Topological characterizations of various ideas from simple games are provided, as are the expressions for Banzhaf and Shapley-Shubik power indices for weighted games. We calculate the indices in several examples of weighted voting games with unfeasible coalitions, including the U.S. Electoral College and the Parliament of Bosnia-Herzegovina.