A Complete Geometric Representation of Four-Player Weighted Voting Systems

The relatively new weighted voting theory applies to many important organizations such as the United States Electoral College and the International Monetary Fund. Various power indexes are used to establish a relationship between weights and influence; in 1965, the Banzhaf Power Index was used to show that areas of Nassau County were unrepresented in the county legislature. It is of interest to enumerate weighted voting systems, analyze paradoxes, and solve the “inverse problem” of constructing a voting system from a desired power distribution. These problems are usually addressed using the standard algebraic representation of weighted voting games consisting of a weight vector and a quota. Other ways of representing weighted voting games do exist, such as the set of minimum winning coalitions, an idea addressed in several papers. A newer idea, however, is the geometric representation. This representation contains all possible normalized n-player weighted voting games in a (n− 1)-simplex and thus acts as a complete representation of weighted voting games. The concept of the region, a portion of the simplex producing characteristically identical weighted voting systems, may greatly simplify analysis of weighted voting games. In this paper, four-player weighted voting games are completely solved using the geometric representation. The geometric representation will be shown to be a useful alternative to the algebraic representation.