The topology of simple games
Ismar Volic, Leah Valentiner
TL;DR
This paper forges a bridge between simple game theory and combinatorial topology by identifying losing coalitions with a simplicial complex $K_{\mathcal{L}}$ on $N$. It builds a dictionary translating notions such as duality, sums/products, and restrictions into topological constructions (Alexander duality, joins, cones, maps) and invariants (homology), and proves a key homology-based characterization of symmetric weighted games: a simple game with $n$ players and quota $q$ is symmetric iff $\beta_i(K_{\mathcal{L}}) = {n \choose q}$ for $i=q-2$ and $0$ otherwise. The results recover known game-theoretic facts via topology and provide new insights, such as homology-based criteria for weighted-symmetric cases. The work lays a foundation for further exploration, including category-theoretic formulations and hypergraph extensions, connecting game theory with topology in potentially fruitful ways.
Abstract
We initiate the study of simple games from the point of view of combinatorial topology. The starting premise is that the losing coalitions of a simple game can be identified with a simplicial complex. Various topological constructions and results from the theory of simplicial complexes then carry over to the setting of simple games. Examples are cone, join, and the Alexander dual, each of which have interpretations as familiar game-theoretic objects. We also provide some new topological results about simple games, most notably in applications of homology of simplicial complexes to weighted simple games. The exposition is introductory and largely self-contained, intended to inspire further work and point to what appears to be a wealth of potentially fruitful directions of investigation bridging game theory and topology.