Introduction and Outline
It’s just the greatest number of votes wins the election, right? Not so fast.
While a simple counting of votes can be an effective method in some situations, the theory of designing voting methods is a lot more complicated (and intriguing) than that. When one asks, “which voting method is the best,” this is both a mathematical and a philosophical question, not to mention a question that lends itself to interesting pedagogical opportunities in a math classroom. In this article I’m going to break this question into many pieces, but let’s start with defining the phrase “voting method.” A voting method is a procedure used on a preference table to determine a winning party out of two or more parties running in an election.
Now then, what’s a preference table? A preference table is a numerical table that indicates how many people voting in an election rank all of the candidates in a certain way. The table will have one column for each possible order of ranking the candidates. Since most examples used in a classroom setting have 3 candidates running in the election, I will indicate now that a preference table in this situation will have 6 columns. That’s because we can rank three candidates A, B, C in 6 ways:
A > B > C
A > C > B
C > A > B
C > B > A
B > A > C
B > C > A
The meat of this article will be using one preference table to demonstrate how four separate voting methods can be put to work, so rest assured if you’re wondering what a preference table looks like. We’ll explore some of the nuances and subtleties of voting methods, and discuss some advantages for including this content in the teaching of a collegiate recreational mathematics course.
Overview of Four Voting Methods
Let’s break down four common voting methods that will be utilized and picked apart as we move along. Each method describes a procedure by which votes are counted or scored, and then a winner is determined. All of the methods presented here are deterministic. That means there is no chance for a random event to produce multiple different winners. Non-deterministic voting methods can be generated and may be closer representations of how voting takes place in the real world, but the result would be too complicated to teach in an entry-level mathematics class.
- Plurality: each person voting in the election ranks all of the parties based on an order of preference. The party with the greatest number of first place preferences wins.
- Plurality with elimination: just like in the plurality method, count the number of first place preferences for each party. If one party has over half of the first place preferences, that party is the winner. Otherwise, completely remove the lowest ranking party from the election (and the preference table) and tally the votes again.
- 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.
- Pairwise Comparison: run several smaller elections, each featuring a one-on-one matchup (using the original data in your preference table). Whichever party wins the most one-on-one matchups is deemed the overall winner.
Especially with the plurality method, it is easy to see that it is possible for these voting methods to wind up with a tie, in which case some other means must be used to determine a single winner.
Here’s the real kicker, though: even with the same preference table, using a different voting method can result in a different winner. This makes it difficult to say that one voting method is universally better than another. To partially rectify this, we introduce some fairness criteria towards the end of the article, that will serve as one way to compare the different voting methods to each other.
Detailed Example: Voting Methods in Action
Now it’s time to put our four voting methods on a grand display and show how the procedures work. For maximum coherence we’ll use the same preference table for all of the examples. Doing this also reveals the possibility that different voting methods used on the same preference table will yield different outcomes.
For this section, the hypothetical scenario is as follows: I surveyed 50 friends and asked them to rank their preference between three pizza toppings: pepperoni, mushrooms, and banana peppers (the latter is a bit untraditional but it’s one of my personal favorites!).
The preference table is as follows:
Notice that there are six possible ways to rank the three toppings, so there are six columns in the table (not including the header column on the left). To make the table’s contents more concise, I abbreviate P = pepperoni, M = mushroom, and BP = banana peppers. Then we can read the table column by column. For instance, the column with a “13” at the top means that 13 (out of 50) people surveyed said they rank pepperoni as their first choice topping, mushrooms as their second choice, and banana peppers as their third choice. The five columns that follow can be read in the same exact way. It’s also worth double-checking at this point that the numbers running across the first row of the table add up to 50. No person voted more than once, and everybody that participated in the survey did cast a vote.
Now we’ll indicate how to utilize each voting method on this preference table.
Plurality: for this method, only the first row of the table (the “first choice” row) is of any relevance. Each pizza topping appears twice in this row, and so we can find how many people voted for each topping as their first choice preference:
1st choice for pepperoni: 13 + 6 = 19 people
1st choice for mushrooms: 2 + 7 = 9 people
1st choice for banana peppers: 14 + 8 = 22 people
Since banana peppers got the greatest number of first choice votes, banana peppers win the election when plurality is used. As a sanity check, notice that
19 + 9 + 22 = 50
This makes sense because 50 people voted, and each person voted exactly once. Performing this quick calculation can help you check your work.
Plurality with elimination: the nice thing about this method is that it builds upon the work we did to determine the winner using plurality. In particular, the first step of both methods is to count the number of people who picked each pizza topping as their first preference. To recap,
1st choice for pepperoni: 19 people
1st choice for mushrooms: 9 people
1st choice for banana peppers: 22 people
The next step for this method is to see whether the most popular first choice carried more than half of all the votes. In this case, the most popular choice netted only 22/50, or 44% of all the votes. That means we have to eliminate the least popular choice and count the votes again. As you can see from the numbers above, that choice is the lowly mushroom. Let’s cross out the mushroom wherever it appears in our original table:
Now we can revise the table to only include people’s preferences between pepperoni and banana peppers. That means there will be one less row in the table. Here it is:
Notice that some of the preferences “slid upward.” The first column originally had people who put mushroom as their second choice preference, but that is no longer an option. Hence, for those people, banana peppers were promoted from third choice to second choice. This same pattern showed itself in the last three columns as well.
There are two options left, and only one winner. We find that winner by tallying the number of first choice votes for pepperoni and for banana peppers:
Pepperoni: 13 + 6 + 7 = 26
Banana peppers: 14 + 8 + 2 = 24
While it might be a very close race, it turns out pepperoni becomes the winner with this voting method. Unlike the plurality method, where the banana pepper was the winner, the last column becomes very important. Seven people originally voted for mushrooms first, pepperoni second, and banana peppers third. When we ruled out the mushrooms in the elimination, those seven people magically began casting votes for pepperoni, allowing the salty meat topping to clinch an unheard of victory.
Borda count: of the four methods we’re going over, this one is the most time-consuming to execute because the arithmetic is a bit more complicated than for some of the other methods. Here’s the original preference table again:
In Borda count, each first choice awards 3 points, each second choice awards 2 points, and each third choice awards 1 point. These numbers have to be adjusted if there are more than three choices, but that would make our example unnecessarily complicated. In any case, the greatest number of points wins. The score for pepperoni will be three times the number of first choices, plus two times the number of second choices, plus one times the number of third choices. With the numbers above, the formula is
13(3) + 6(3) + 14(2) + 8(1) + 2(1) + 7(2) = 39 + 18 + 28 + 8 + 2 + 14 = 109
In the same exact fashion, we can find the score for mushrooms:
13(2) + 6(1) + 14(1) + 8(2) + 2(3) + 7(3) = 26 + 6 + 14 + 16 + 6 + 21 = 89
Finally, let’s find the score for banana peppers:
13(1) + 6(2) + 14(3) + 8(3) + 2(2) + 7(1) = 13 + 12 + 42 + 24 + 4 + 7 = 102
Of these scores, pepperoni got the highest, so we say that pepperoni wins by borda count.
Pairwise comparison: in this last voting method, we need to carry out three mini-elections, and determine whether one pizza topping prevails amongst these mini-elections. As we said before, the condition to do that is winning two of them. The three pairs we need to look at are:
- Pepperoni vs. mushroom
- Pepperoni vs. banana peppers
- Mushroom vs. banana peppers
To make it easier to keep track of the information relevant for each mini-election, it helps to highlight certain portions of the preference table. Here’s what the highlighted table looks like for pepperoni vs. mushroom:
In each column, we care about which highlighted pizza topping appears closer to the top of the table. That’s the topping that gets the points in that column. Put more explicitly:
In the first, second, and third columns, pepperoni is above mushroom, so pepperoni gets 13 + 6 + 14 = 33 votes.
In the fourth, fifth, and sixth columns, mushroom is above pepperoni, so mushroom gets 8 + 2 + 7 = 17 votes.
From this, it is clear that pepperoni wins this mini-election.
What does the mini-election for pepperoni vs. banana peppers look like? Well, let’s start with another highlighted preference table.
In the first, second, and sixth columns, pepperoni is above banana peppers, so pepperoni gets 13 + 6 + 7 = 26 votes.
In the third, fourth, and fifth columns, banana peppers is above pepperoni, so banana peppers gets 14 + 8 + 2 = 24 votes.
Pepperoni wins this mini-election, but it has already won two mini-elections, so we know pepperoni is the overall winner by pairwise comparison. To be complete, we still detail how the last mini-election is carried out. It is mushroom vs. banana peppers:
In the first, fifth, and sixth columns, mushroom is above banana peppers, so mushroom gets 13 + 2 + 7 = 22 votes.
In the second, third, and fourth columns, banana peppers is above mushrooms, so banana peppers gets 6 + 14 + 8 = 28 votes.
The Four Fairness Principles
There are four main criteria through which we measure whether a voting method is truly fair, or reasonable. Of course, there are some questions centered around whether these are the right criteria to consider, but we’ll defer that discussion to the next section. For now let’s list the criteria:
- 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”
Now here are some comments on how to interpret, or break down, each of these principles.
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.
Condorcet: this criterion is most easily depicted with an example election involving three parties. Let’s call the parties A, B, and C. Then the condorcet criteria works as follows: run three “mini-elections:” A vs. B, A vs. C, and B vs. C. Each party participates in two of the three mini-elections. If a party manages to win both of the mini-elections they participate in, then the condorcet criteria says that party should win the overall election. In some cases there would be no party who wins two mini-elections; suppose for instance A beats B, C beats A, and B beats C. Here, the condorcet criterion is inconclusive.
Monotonicity: The name for this fairness criterion is inspired by the concept of a monotone function. A monotone function f can be either [monotone] increasing or [monotone] decreasing. If it’s increasing then that means f(a) < f(b) whenever a < b. Similarly, a decreasing function will have the property f(a) > f(b) whenever a < b. Either type of function may be referred to as monotone. So how does this help us? Well, rather than dealing with an algebraic function, we think of the monotonicity fairness criterion as an analogy to algebraic functions. This criterion says that the voting scenario behaves kind of like a monotone function. So if one person wins the election, and then the votes are modified so that more votes go to the winner, than that person (or party) should still win.
Independence: This name for a fairness criterion basically indicates that the winner is independent of losers. Removing losers from the election shouldn’t change the winner. In some textbooks, this term is referred to by its full name, Independence of Irrelevant Alternatives, abbreviated IIA.
That’s how I would interpret the four fairness principles. They can also be treated as mnemonics to keep track of which one is which. Might be helpful if you need to know these definitions for a test. Anyway, let’s discuss the fairness principles from a philosophical standpoint.
Philosophical Discussion Behind the Fairness Principles
It may seem that the four fairness criteria outlined earlier were chosen arbitrarily. You might have questions floating around in your head such as, “what does it even mean to be fair?” Many people will start listing synonyms of fair that are used in special contexts such as in discussion of games or the law. To be frank, there isn’t a single right answer to a question like this. The fairness principles just seem to be reasonable because they are synonymous with common sense, even outside of a voting context. They are also relatively simple to explain in non-mathematical terms that a layman could understand, which makes them attractive choices. On the flip side, that means it is a relatively simple task to identify when one of those principles is violated, for a given preference table. A complete treatment of voting methods in a math classroom will certainly include examples of preference tables that reveal when a voting method is shown to fail to meet a fairness criteria. In some contexts, it may make sense to choose a voting method that forgoes one fairness criteria to ensure another is met.
There are also some pedagogical advantages to discussing the concept of fairness in a math classroom. The main boon is that such a discussion can be refreshing and invoke different trains of thought than might otherwise be encouraged. In particular, mathematics is a fact-based subject, where facts are proven, but the concept of fairness sprouts questions that are based more so on opinions. Students may even be able to draw on personal experiences or current events to support their arguments. When a curriculum permits, I encourage instructors to facilitate discussions on these issues and give students a chance to connect course material to things they may have seen or heard elsewhere. This is especially important in light of recent events such as the 2020 U.S. presidential election and the Capitol raid that took place two weeks ago (prior to the publication of this article).
Arrow’s Impossibility Theorem
As a concluding thought for the philosophical discussion on the fairness criteria, we present a useful fact that acts as a nail in the coffin. This is a philosophical and mathematical principle known as Arrow’s Impossibility Theorem. It states the following:
When voters have three or more options, there is no possible ranked electoral voting system that is guaranteed to satisfy all four criteria of fairness (majority, condorcet, monotonicity, independence)
While we won’t talk about the proof of this theorem in this article, there is an important takeaway from it: no voting system is perfect, and all of the systems we discussed have inherent problems that can arise for specific preference tables. If you want to understand the voting methods in greater depth, my suggestion is to build lots of preference tables and carry out all of the voting methods we discussed. Then, determine if any of the fairness criteria we analyzed are violated. Designing other voting methods of your own is also a meaningful exercise.
Notice that of the voting methods and fairness principles listed here, one could argue that pairwise comparison is the “most fair” because it satisfies three of our four fairness criteria. Of course, the definition of “fair” is a little bit up in the air.
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.