An Introduction to the Math of Voting Methods

By Brendan W Sullivan

Some modern political discussions are focused on electoral reform and the mechanics of democracy. For instance, Maine and Alaska recently adopted new procedures for statewide elections that involve Ranked Choice Voting, while a similar ballot measure in Massachusetts was only narrowly defeated. Meanwhile, co

• Language: English
• Publisher: 619 Wreath
• Release date: Oct 3, 2022
• ISBN: 9781958469057

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Majority Voting and Plurality Voting (First Past the Post)

The goal of this chapter is to introduce you to Plurality Voting (also known as First Past the Post ), which is common and popular around the world. We will describe how it works and where it is used, and briefly discuss some of its pros and cons. Along the way, we will introduce some important concepts and examples that will be mentioned throughout the rest of the book.

1.1

Definitions and Examples

Let’s begin with the simplest voting method, or set of rules for calculating a winner of an election.

Definition 1.1: Majority Rules Voting.

This is a voting method that only applies when there are two candidates. Each voter chooses one of the two candidates. The winner is the one that receives the most votes, which must be a majority (more than half, > 50%).

If there is an even number of voters, then it is possible for the result to be a perfect tie, so the organizers of a real-world election may wish to have a tie-breaker rule in place. However, this is not a mathematical issue: it depends on the context. How to handle a tie in an election for a Senate seat might be very different from how you handle a tie in a friendly vote about what to have for dinner! (And with thousands or even millions of voters in a Senate race, a perfect tie would be astronomically unlikely, anyway.) So, when we look at examples in this book, we will not deal with ties all that often, and we will never presume any particular method for breaking a tie.

Example 1.1. Suppose we have nine friends who want to order a meal for a group study session. In a friendly discussion, they narrow down their options to Pizza or Indian food. They agree that everyone will write down either P or I on a slip of paper to indicate their choice, and then the Majority Rules method will be used to decide the winner.

Let’s say that five friends chose P and the other four chose I. In that case, Pizza wins the election with 5 votes out of 9 total votes, or a proportion of ⁵⁄9, which is approximately 55.6%.

This simple example brings up a few important ideas:

In this example, the candidates are the options to choose from: Pizza and Indian food. In some examples, the candidates will be people running for political office, but this example shows that this is not always the case.

In this example, the voters are the nine friends. In some examples, the voters may not even be people, but usually they will be.

In any example you come across, in this book or elsewhere, it is probably worth spending a moment to think about who/what the candidates are, and who/what the voters are.

In this example, a ballot submitted by one of the voters is just an indication of which option is that voter’s top choice. Imagine that you’re one of the friends and you want Indian food. By writing I on your slip of paper, you are indicating that as your top choice.

If you had instead written "1st: I, 2nd: P ," that would provide the exact same information, but with more writing. In other words, with only two candidates, you can just indicate which one is your 1st place choice; it is automatically implied that the other option is your 2nd place choice. This may seem silly to point out now, but very soon we will look at situations with three or more candidates, in which case a voter’s ballot may contain much more information than merely saying, Here is my #1 choice.

Beyond what the ballots say, it is also worth considering how the ballots are submitted. What if the friends raised their hands to vote publicly? Perhaps some of them would jump on the bandwagon and vote for the popular choice because they see almost everyone else raising their hand already. Or, in a larger group, perhaps they would use some kind of online survey to gather the votes instead of using slips of paper.

These are important considerations if you’re going to hold an election or participate in one. However, you can see that they are not really mathematical issues; rather, they depend on the real world context of the election. For this reason, we will not really discuss such issues in this book. For the most part, we will focus on how to take the information on the ballots and calculate the winning candidate. All other considerations (how the candidates are chosen, what voters get to participate, how the ballots are submitted, etc.) are outside of our scope.

Let’s now look at our first example of a voting method that applies when there are more than two candidates. Imagine the friends decide to expand their meal options: Pizza, Indian food, and Thai food. In this situation, majority rules may not apply because the votes can be split among the three options, causing none of them to have > 50% support. This is what we will see in the example below, after the definition.

Definition 1.2: Plurality Voting.

This is a voting method that applies when there are three or more candidates. Each voter chooses one of the candidates. The winner is the one that receives the most votes. It is possible for the winner to have less than half of all the votes, in which case their share is called a plurality. (Remember that the term majority is only used when the winner has > 50% of all the votes.)

Example 1.2. Suppose the nine friends decide that each of them will write I (Indian food) or P (Pizza) or T (Thai food) on a slip of paper. The votes will be counted, and the option with the most votes will be the winner. Let’s say four friends choose I, three choose P , and two choose T . In other words,

Indian food gets ⁴⁄9 of the votes, or approximately 44.4%.

Pizza gets ³⁄9 of the votes, or approximately 33.3%.

Thai food gets ²⁄9 of the votes, or approximately 22.2%.

You can see that Indian food is declared the winner with a mere plurality (44.4%) and not a majority (> 50%).

A few comments about this example:

In the previous example with only two options, Pizza got five votes. In this new example here, it may help to imagine that two of the friends switched their choice from Pizza to Thai food when that option became available, while all four of the voters who picked Indian food originally stayed with that choice.

This is just one example of a winner having a plurality but not a majority. We will see many examples of that in this book, and we will mention plenty of similar real world examples.

We specifically chose the same underlying situation to use in these first two examples to make it clear that Majority Rules is Plurality Voting when there are only two candidates. In other words, these two methods are fundamentally identical. The only differences arise when the number of candidates is more than two.

1.2

Real World Usage

Plurality Voting is used all over the place; it’s fair to say it’s the most popular voting method, at least by frequency of use. If you have participated in pretty much any election in the United States, then you have used Plurality Voting.

You may have seen the phrase First Past the Post used, perhaps in a news report about voting. This phrase refers specifically to an election where Plurality Voting is used and there is a single winner. For example, every two years in the US, you get to vote for your representative in the U.S. House of Representatives. There may be several candidates, but you only get to indicate your support for one of the candidates on the ballot, and the single winner is the one who gets the most support.

Many nations around the world use a First Past the Post (Plurality) method when citizens vote for their representatives in the national legislature. This includes the U.S. (Senate and House of Representatives), the United Kingdom (Parliament), Brazil (Federal Senate), Canada (House of Commons), Azerbaijan (National Assembly), Ghana (Parliament), India (Lok Sabha, or House of the People), and many more. See the Wikipedia page on First-past-the-post voting for a list of countries that currently use such a method to elect their Head of State or National Legislature. [3]

You have likely also encountered Majority Rules Voting and Plurality Voting in school. Perhaps a teacher needed to decide what to do next, and they held a vote among the class. If there were only two options, they probably used the Majority Rules method. And if there were three or more options, they probably used the Plurality Voting method. We’re guessing that this applies to your Student Government elections, too: voters get to pick one of the options (however many there are), and the winner is the one who gets the most votes.

In short, Plurality Voting is used all over the place because it is straightforward and simple to implement. However many candidates there are, each voter just picks one and indicates that choice somehow. Then, those choices are counted and you see which total is largest. Couldn’t be simpler, right? This is one of the pros of the method that we will discuss shortly.

1.2.1

Real World Examples of Elections

Let’s look at a couple of specific examples of political elections in the U.S. that used Plurality Voting. They have been chosen to point out some important ideas and contribute to some later discussions.

Example 1.3 (Massachusetts’ 3rd Congressional District, 2018). The general election for all U.S. House of Representatives seats was held on November 6, 2018. In Massachusetts’ 3rd District, there were three main candidates; here are their vote totals [4]:

Lori Trahan (Democrat) won 173,175 votes, about 62% of the total.

Rick Green (Republican) won 93,445 votes, about 33.5% of the total.

Mike Mullen (Independent) won 12,572 votes, about 4.5% of the total.

There were also 135 write-in votes for other candidates.

You can see that Lori Trahan won the election and she even had a majority of the votes. However, there was a much closer race two months earlier: the Democratic primary was held on September 4, but the results were so tight that a manual recount didn’t conclude until two weeks later.

In that primary, Lori Trahan won with not a majority but rather a very slim plurality. There were ten candidates all vying for the Democratic spot on the ticket in the upcoming general election. The votes were pretty well spread across all ten candidates so that no one got even close to a majority. In fact, five candidates all earned somewhere between 15% and 22% of the votes, accounting for almost 90% of all votes (with the other five candidates accounting for the remaining 10% or so). Here are the results for the top six [4]:

Lori Trahan won 18,580 votes, about 21.7% of the total.

Dan Koh won 18,435 votes, about 21.5% of the total.

Barbara L’Italien won 13,018 votes, about 15.21% of the total.

Juana Matias won 12,993 votes, about 15.18% of the total.

Rufus Gifford won 12,873 votes, about 15.04% of the total.

Alexandra Chandler won 4,846 votes, about 5.7% of the total.

Wow! It’s remarkable how close 1st and 2nd place were, as well as how close 3rd, 4th, and 5th all were. But more importantly, it’s worth considering the fact that while 21.7% of the voters in that district chose Lori Trahan, 72.3% of the voters chose someone else. With 10 candidates, the votes can be spread out so much that the winner receives a rather small percentage of the total.

Now, we don’t mean to cast doubt on Lori Trahan or the legitimacy of the election, not at all. Presumably, many of the the voters who chose someone other than her were totally fine with her winning. But what if a lot of them really hated her and would have ranked her last, if they could have put all the candidates in their order of preference? Wouldn’t that be strange, to have such a widely unpopular candidate eke out a victory, and partly just because there were so many people running? Again, we are not saying this is actually what happened in that election, but it is important to recognize that such a thing could easily happen, because of the rules of the Plurality Voting method.

1.2.2

Moving Away from First Past the Post

Because a major goal of this book is to give you information with which to have an informed discussion about different voting methods, it’s also worth mentioning now that several countries have recently changed from First Past the Post voting to something else.

For example, in 2003 Papua New Guinea changed from First Past the Post to something similar to Instant Runoff Voting (which we will learn about in the next chapter, and is also known as Ranked Choice Voting). Even more recently, Lebanon moved to a proportional representation system in 2018. Each district is represented by multiple members of the legislature, and voters in a district can vote for several candidates. The results of that district are based on the proportion of all votes that each candidate receives.

In the United States, Plurality Voting is still used in all elections for Congress and the President (although the Electoral College complicates the process somewhat), except in Maine and Alaska. In 2016, voters in Maine narrowly approved a ballot referendum that stipulates Ranked Choice Voting will be used in all Congressional elections from now on. The midterm election in 2018 was the first time a method other than Plurality Voting was used in a statewide election in the US. And voters in Alaska passed a ballot referendum in 2020 that will see them using a two-round combination of Plurality Voting and Ranked Choice Voting in the future. (All of this will be discussed later in Section 2.2.2.0 .)

For now, we hope that you see that Plurality Voting is common and widespread, but also that it does not have to be so common. Let us now discuss some reasons for and against using this voting method.

1.3

Discussion of Pros and Cons

1.3.1

Pros

We’ve already mentioned the most obvious pro of Plurality Voting: simplicity. It’s worth describing how this applies in a few different ways:

Simplicity for the voters. The instructions on a ballot are easy to state: select one. And those instructions are easy to follow: a voter fills in one bubble, or checks one box, or something like that. In other words, this method has a very low potential for voter error or confusion.

Simplicity in calculating results. Because each submitted ballot essentially has one piece of information (the one selected candidate), it’s straightforward to count how many votes each candidate receives. This can become logistically challenging when there are millions of voters, but this method is, at least, mathematically very simple.

Simplicity for the candidates. Assume we’re talking about a political race. When each candidate knows that the most votes wins and that’s it, they will act accordingly in their campaigns. They will tell you (the voters) why to choose them and not the others. Although this is not a great system and tends to encourage negative campaigning, this is simple.

1.3.2

Cons

Although it may seem strange, one of the most popular arguments about the downside of