The topology of simple games

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 which have interpretations as familiar game-theoretic notions and objects such as the arithmetic of games and dual games. We also provide some new topological results about simple games, most notably in applications of homology of simplicial complexes to weighted simple games. The main result along these lines is the characterization of symmetric games using homological Betti numbers. 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.