A Complete Geometric Representation of Four-Player Weighted Voting Systems

This paper seeks to expand voting power theory, a branch of game theory that applies to many important organizations. Typically, weighted voting systems are displayed using the algebraic representation, consisting of a quota and a weight vector. A newer idea, however, is the emph{geometric representation}. This representation maps all normalized weighted voting systems onto a simplex and thus can be called a complete representation of weighted voting systems. The concept of the emph{region}, sets of characteristically identical weighted voting systems, will be introduced, greatly simplifying the analysis of weighted voting systems. In this paper, four-player weighted voting systems are solved completely using the geometric representation. The geometric representation will be shown to be a useful alternative to the algebraic representation