Voting Theory

With Peri Shereen and Jeffrey Wand

One of the applications of game theory, linear algebra, combinatorics, and, more recently, representation theory is voting theory. In particular, realizing a voting system as a game allows one to get more information about the complexity of said voting system.

A recent paper by Cheung and Ng explains the process of calculating the “dimension” of a voting system in Hong Kong. The dimension of a voting system measures how complex the voting system is. For example, the passage of a Bill in the US Congress, which is determined by a majority vote, has a dimension of 1. There are also examples of voting systems of dimension 2, but until more recently, examples of voting systems of dimension 3 or higher were not known, (explained in a paper by Tyler). Thus, the main goal of the paper by Cheung and Ng is to prove that the voting system in Hong Kong has dimension 3, a non­trivial task.

Our proposal is to start by introducing the ideas of game theory, combinatorics, and linear algebra needed to read the paper by Cheung and Ng. Then we continue with an easier example of dimension by proving that the US congressional voting system has dimension 1. Next we go through Cheung and Ng’s paper. Finally we look for other examples of higher dimensional voting systems in other countries, (maybe even find ones with dimensions higher than 3). This project is open to juniors and seniors.