Voting Theory

With Peri Shereen and Jeffrey Wand

A normalized weighted voting system can be mapped onto a simplex where each point represents a weight distribution for each player. For example, a normalized game with three players can be mapped onto an equilateral triangle whose vertices lie on the points (0,0,1), (0,1,0), and (0,0,1). Viewing voting systems in this way allows us to investigate some of voting's paradoxes. One such paradox occurs with a redistribution of weight. It is possible for a player to lose some of their weight, but gain power and vice versa. Geometrically what happens with such a redistribution is the "passing through of a hyperplane" of our simplex. Our goal is to investigate such paradoxes using this geometry and try to find realizations for higher dimensional voting systems.

We will start our project by learning about simple weighted voting systems, higher dimensional voting systems, and power indices for players (in particular we will look at Banzhaf and Shapley-Shubik indices). We will then read through Jones' paper "the Geometry behind Paradoxes of Voting Power." Lastly, we will see if we can generalize the notion of power indices to voting systems whose dimension is greater that 1. For this project it will be useful to be familiar with concepts though Math 10A.