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:
- Majority: “getting a majority (50+%) of the votes should guarantee a win”
- Condorcet: “if one wins over each of the others when paired up, then it should win overall”
- Monotonicity: “if one wins, and then if there is a re-election, if all changes favor that one, then that one should still win”
- Independence: “if one wins, and then non-winners are removed from the election, then that one should still win”
The criterion that will be most relevant for the discussion in this article will be the majority(the other ones are discussed in more detail here).
Majority: count all of the votes, and any individual who has more than half of the votes should be the guaranteed winner. Adding up all the votes for all people running in the election will give the total number of votes. Also, it’s worth noting that only one person can carry more than half of the vote, and that there won’t be a person in every election with a majority.
Oh, and our example in this article will involve 50 people casting votes for their favorite pizza topping, between pepperoni (“P”), mushrooms (“M”), and banana peppers (“BP”).
Violate Majority Criterion via Borda Count
In this section we will look to violate the majority criterion when the voting method of choice is the Borda Count. That means we want to find a preference table where one party gets half the total number of first-place votes, but doesn’t win if the election is done with Borda Count. For the examples in this article, 50 people are participating in the election, so we will aim to use a preference table where one party (a.k.a a pizza topping) gets at least 25 of the first-place votes, but doesn’t win a Borda Count election. We’ll provide an example where this happens, and then explain heuristically why it’s a sensible example to construct.
It is very quick to see that pepperoni garners the majority of the first-place votes, or 26 out of 50, which is more than half. For determining the majority, it’s irrelevant how many first-place votes the other two toppings received. Anyway, our goal now is to see which topping wins with the Borda count protocol. To recap, a first-place vote scores three points; a second-place vote scores two points; and finally a third-place vote scores one point.
We score each topping one at a time…
Pepperoni: 26(3) + 5(2) + 19(1) = 107
Mushroom: 20(3) + 24(2) + 6(1) = 114
Banana pepper: 4(3) + 21(2) + 25(1) = 79
From these tallies it appears that mushrooms win the Borda count. Since pepperoni had the majority of voters listing it as their favorite topping, this preference table is an example of where Borda count can violate the majority criterion. But that example wasn’t pulled out of thin air! There was a bit of work done in selecting this particular preference table.
Analysis: The key to why this preference table works is that mushrooms got a large portion of the second-place votes. Second-place votes are irrelevant when determining a favorite topping by majority, but they are very important when doing Borda count. At the same time, it’s important to minimize the number of points pepperoni gets beyond those people awarding it 3 points each. That means a preference table that accomplishes the intended goal will have a lot of people placing mushroom as second place, and as many as possible placing pepperoni as third place. Of course, the majority of people can’t put pepperoni as third place, since the majority already said it was their favorite. So we just make the number of people ranking pepperoni second small.
Clark and Clark, “The Beautility of Math” published by Great River Learning
This textbook is used at the University of Tennessee for the entry-level mathematics course “Mathematical Reasoning,” and the content of this blog article was in part based on this text. I use the aforementioned text when I teach said course myself.