Violating the Majority Criterion

Introduction and outline

In this article I’m going to describe some mechanisms for carrying out an election that are mathematical and deterministic in nature (i.e. noting happens randomly). These mechanisms are called voting methods, because they are essentially ways of running an election to determine a winner. The data used in these elections come in the form of preference tables. While there are many voting methods studied in research papers and utilized as examples in recreational mathematics classes, this article will one particular voting method as a case study:

Borda Count: each person voting in the election ranks all of the parties based on an order of preference. The parties higher up in the preference order score more points, and the parties lower down the preference order score less. Each party gets a score, and the highest score wins.

There are a lot of mathematical nuances involved with voting methods, and we will try to explore some of those in this article, particularly through the lens of fairness principles. A fairness principle is a means of determining whether a voting method is reasonable from some ethical or moral standpoint. These principles are somewhat vague in terms of whether they satisfy certain philosophical trains of thought, but we won’t get into that here. Here are four fairness principles that are commonly studied in juxtaposition with voting methods:

1. Majority: “getting a majority (50+%) of the votes should guarantee a win”