Integrating Combinatorics, Geometry, and Probability through the Shapley-Shubik Power Index

Abstract

Introduction It is our belief that students compartmentalize mathematical techniques to be used solely for a specific problem or narrow set of problems. Ideally, students would develop a toolbox of mathematical techniques to analyze any problem from a multitude of perspectives. Analyzing simple weighted-voting games helps students develop varied approaches to problem solving while demonstrating how to use different mathematical skills in nontrivial, relevant ways. Such an analysis necessitates the integration of mathematical topics, including combinatorics, geometry, and probability. This article serves as a primer for instructors so that they may introduce simple weighted-voting games and the Shapley-Shubik power index in order to relate voting theory to various topics in the curriculum. To encourage implementation and adaptation of this material, we include many examples and exercises. For this reason, this article may be used for self study by independent study students. These materials have been developed in a handful of courses at Montclair State University from spring 1999 to the present, including a general education requirement course, an upper level applied combinatorics and graph theory course, a graduate level course in combinatorial mathematics, and two independent study courses. One student has used this article as a self-study guide as a precursor to computational voting theory. These different levels of use are a testament to the diversity of mathematics that can be used to analyze simple weightedvoting games through the Shapley-Shubik power index.