The topology of simple games (2503.12578v1)

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.

An Exploration of Simple Games Through Combinatorial Topology

"The Topology of Simple Games" by Leah Valentiner and Ismar Volić embarks on a comprehensive examination of simple games through the lens of combinatorial topology. By modeling losing coalitions as simplicial complexes, the paper aims to establish a novel interaction between game theory and topology, potentially revealing new insights and methodologies applicable to both fields.

The paper focuses on simple games, a foundational class of cooperative games characterized by a binary outcome that dichotomizes coalitions into winning or losing. These games are reminiscent of voting scenarios where coalitions represent groups of players cooperating to achieve a common decision outcome, such as legislative votes. The researchers uniquely leverage the structure of simplicial complexes—a core concept in combinatorial topology—to represent losing coalitions, thereby facilitating the application of topological principles to game theory.

Correspondence and Topological Interpretations

Valentiner and Volić highlight key correspondences between simple games and simplicial complexes. The components of losing coalitions align directly with the facets of simplicial complexes. The authors elucidate critical game-theoretic constructs, such as dummy players, dual games, and homomorphisms, with topological equivalents such as cones, Alexander duals, and simplicial maps.

The paper's rigorous exploration reveals that the combination of two simple games, via sum or product operations, corresponds topologically to the join and intersection of simplicial complexes in which the losing coalitions reside. These insights offer deep topological interpretations of standard game-theoretic operations and contribute to an enriched understanding of coalition structures.

Homology and Symmetric Weighted Games

A significant contribution of the paper is the application of homology, a robust invariant in topology, to the framework of simple games. The researchers determine criteria for symmetric weighted games using homological constructs, revealing that the homology groups can describe the symmetric nature of weighted voting systems. The precise determination of which coalitions are winning or losing is equated to configurations in the homological structure of the simplicial complexes.

This utilization of homology provides a concrete numerical framework that not only carries theoretical implications but could also guide practical decision-making processes in weighted voting contexts seen in entities like the United Nations or the European Parliament.

Beyond Simplicial Complexes: Hypergraphs and Compatibility

The paper also discusses the relevance of hypergraphs, which allow for modeling winning coalitions without downward closure requirements. While recognizing the complexity introduced by hypergraphs, the authors propose the potential of hypergraph models for examining structures and dynamics within more general game-theoretic scenarios.

Moreover, Valentiner and Volić suggest the potential of modeling real-world coalition behaviors wherein certain player subsets are incompatible, which they address by defining "observable coalitions," showcasing the comprehensive adaptability and versatility of their approach.

Future Directions and Broader Implications

"The Topology of Simple Games" opens a myriad of avenues for further research. By interfacing game theory with the established mathematical frameworks of algebraic topology, the authors invite explorations into understanding the deeper topological structures underlying voting systems and coalition formations. The possibility of defining categories for these mathematical spaces and identifying interactions between topological constructs and game-theoretic operations could usher in revolutionary perspectives on both the theoretical foundation and practical applications of game theory.

In summary, Valentiner and Volić's work bridges a gap between topology and game theory, employing a sophisticated language of simplicial complexes to enrich our understanding of simple games. This innovative cross-field examination paves the way for future scholarly exploration and potential breakthroughs in the conceptualization and application of game theory.