In this section, we will explore the following questions.
How are elections with two candidates different from those with more than two?
What is a plurality, and how is it different than a majority?
How can we fairly evaluate voting systems with multiple candidates?
Subsection7.2.1.1Describing the Problem
Let’s begin with a warmup activity.
Activity7.2.1.
The popular vote totals from the state of Florida in the 2000 U.S. presidential election are given in Table 7.2.2.
Table7.2.2.The Florida popular vote in 2000.
Candidate
Popular Votes
George W. Bush
2,912,790
Al Gore
2,912,254
Ralph Nader
97,488
Others
40,579
In this election, did any candidate receive a majority (more than half) of the popular votes cast in the state of Florida?
Suppose the system used in Florida in 2000 is: whoever gets the most votes wins. Under this system, who wins the presidential popular vote in Florida in 2000? By how many votes does this person win? By what percentage of the vote does this person win?
If George W. Bush and Al Gore had been the only candidates on the ballot in Florida in 2000, do you think that Gore might have possibly received more popular votes than Bush in Florida?
In fact, many political scientists believe that Ralph Nader was a spoiler candidate for Gore in 2000; that is, they believe that a large percentage of the Nader voters likely would have voted for Gore if Nader had not been on the ballot. In this section, we’ll explore the wrinkles introduced by additional candidates, and develop alternative systems for choosing winners of elections. First, we’ll define an important term for a concept you are likely already familiar with.
Definition7.2.3.
A candidate in an election who receives more votes than any of the other candidates is said to receive a plurality of the votes cast.
It is crucial that we understand the difference between plurality and majority.
Discussion7.2.4.
For elections with two candidates, explain why the words plurality and majority mean exactly the same thing.
For elections with more than two candidates, explain why the words plurality and majority do not mean exactly the same thing.
We adopt the following definitions.
Definition7.2.5.
Consider an election with more than two candidates.
Majority rule is the voting system that elects the candidate who receives more than half the votes, if such a candidate exists. If no such candidate exists, the election is declared a tie with no winner.
The plurality method is the voting system that elects the candidate that receives the largest number of votes. The plurality method only produces a tie when two candidates receive exactly the same number of votes, and this number is more than any other candidate.
Discussion7.2.6.
Which of the two methods described in Definition 7.2.5 do you think is more likely to result in a tie?
If a candidate wins an election under majority rule, would that candidate also be guaranteed to win under the plurality method?
If a candidate wins an election under the plurality method, would that candidate also be guaranteed to win under majority rule?
We now explore the plurality method in more depth. Because it is familiar, it likely seems quite fair. But let’s consider the next exploration.
Exploration7.2.7.
It’s time to vote for the mayor of Scranton again, and this time there are thirteen candidates: Jim, Pam, Michael, Dwight, Andy, Angela, Kevin, Stanley, Phyllis, Ryan, Kelly, Meredith, and Toby.
Suppose Kevin wins with 6,000 votes out of 76,328 cast. Did he receive a majority of the votes cast for mayor? (You may want to convince yourself that this is possible!) What percentage of the overall vote did he receive?
What is the smallest number of votes Kevin could have received and still won the election under the plurality method? (Be careful!)
Under the scenario described in the previous question, what is the largest number of voters who could have preferred Kevin least among all 13 candidates and still left him with a chance at winning the election?
Using your answers to the previous question, carefully articulate a critique of the plurality method.
Subsection7.2.1.2A solution
As we saw in Exploration 7.2.7, when there are \(N\) candidates, it is possible for the vote to be split so thoroughly that one candidate can win with just over \(1/N\)-th of the vote. In this case, a majority of the voters will prefer someone other than the person who is ultimately elected. But as we saw, the situation can get even worse than that —it’s possible for the candidate who wins the plurality of the vote to be the last choice of a majority of voters, but still win! The main reason that the plurality method is susceptible to this is that it only accounts for a voter’s first choice; there is no penalty for being a voter’s last choice, and no benefit to being the voter’s second choice.
We will therefore explore methods of voting and fairness criteria that account for a full top-to-bottom ranking of candidates by voters, called a preference order. If there are three candidates in an election, say Michael, Angela, and Stanley, and my preference order is that Michael is my top candidate, Stanley my second choice, and Angela my third, we may write \(M \succ S \succ A\text{,}\) where the symbol \(\succ\) means "is preferred to".
Activity7.2.8.
Consider a 3-candidate election for the mayorship of Scranton between Michael, Angela, and Stanley. How many possible preference orders are there? In other words, in how many different ways could the voters rank them?
Suppose their friend Toby enters the race, bringing the total number of candidates to 4. Now how many preference orders are possible now?
Since our voters will be casting preference ballots, we need a different way of displaying the votes cast.
Exploration7.2.9.
Suppose Stanley, Toby, Angela, and Michael are running for the presidency of Dunder Mifflin, the premier paper company in Scranton. The preference orders for each of the 13 board members of the company are displayed in Table 7.2.10, a visualization known as a preference schedule. The column headings indicate the number of voters with the preference order displayed in the column. For instance, the first column shows that 6 shareholders have the preference order \(S \succ A \succ T \succ M\text{.}\) Note that the preference orders displayed are the five that were cast as preference ballots in the election; there are many others that were possible but not cast, and thus are not displayed.
Table7.2.10.Preference schedule for the presidency of Dunder Mifflin.
Rank
6
3
2
1
1
1
S
T
A
M
A
2
A
A
M
T
T
3
T
M
T
A
M
4
M
S
S
S
S
Under majority rule, what would the outcome of the election be?
Under the plurality method, what would the outcome of the election be?
Rank the candidates based on the outcome produced by the plurality method. The final ranking of candidates by a voting system is known as the societal preference order.
Do you think the plurality winner best represents the will and preferences of the voters? If so, explain why. If not, give a convincing argument for why you think some other candidate would be better.
Subsection7.2.1.3The Borda Count
One very popular method for choosing the winner of a multi-candidate election is the Borda count.
Definition7.2.11.
Consider an election with \(N\) candidates. The Borda count works as follows. Each voter submits a ballot that contains their entire preference order for all candidates in the election. For each ballot cast, points are awarded to each candidate according to the following rules:
A last-place vote is worth 1 point.
A second-to-last-place vote is worth 2 points.
\(\displaystyle \vdots\)
A third-place vote is worth \(N-2\) points.
A second-place vote is worth \(N-1\) points.
A first-place vote is worth \(N\) points.
The candidate who accumulates the largest number of total points from all of the ballots is declared the winner, and the societal preference order is determined by listing the candidates in order of the number of points they received, largest to smallest. If two or more candidates are tied with the largest number of points, they are all declared winners (or some suitable prearranged tiebreaking procedure is used).
Activity7.2.12.
Under the Borda count, what would the outcome of the Dunder Mifflin presidential election in Exploration 7.2.9 be?
Consider the following interesting feature of the Borda count.
Activity7.2.13.
Controversy at the Dunder Mifflin board! Due to some shenanigans involving a major shareholder, the election displayed in Table 7.2.10 has to be rerun. The following preference schedule is produced.
Table7.2.14.Preference schedule for the presidency of Dunder Mifflin.
Rank
6
3
2
1
S
T
A
2
A
A
M
3
T
M
T
4
M
S
S
Who wins under majority rule?
Who wins under the Borda count? Does this seem strange to you?
Lest you think this is a contrived example, be aware that things like this can happen in real life. A version of the Borda count is used by the Associated Press to rank the top 25 college football and basketball teams. In the 1971 AP preseason poll, the author’s Nebraska Cornhuskers received 26 of 50 first-place votes, yet were ranked #2. The results of things like this AP polling anomaly or Activity 7.2.13 suggest a new fairness criterion.
Definition7.2.15.
A voting system satisfies the majority criterion if whenever a candidate is ranked first by a majority of voters, that candidate will be ranked first in the resulting societal preference order.
Discussion7.2.16.
Of all the voting systems we’ve explored thus far, which must always satisfy the majority criterion?
In Section 7.1, we defined three fairness criteria. Our definitions from Definition 7.1.6 can be modified to extend in a natural way to elections with three or more candidates.
Definition7.2.17.
A voting system is anonymous if it treats all of the voters equally, meaning that if any two voters traded preference orders, the outcome of the election (and the resulting societal preference order) would remain the same.
A voting system is neutral if it treats all of the candidates equally, meaning that if every voter switched the positions of two particular candidates in their individual preference orders, the positions of these two candidates would switch in the resulting societal preference order as well.
A voting system is monotone if changes favorable only to a particular candidate in individual preference orders cannot cause that candidate to be ranked lower in the resulting societal preference order.
Exploration7.2.18.
Which of the properties of anonymity, neutrality, and monotonicity are satisfied by plurality? Which are not satisfied? Give a convincing argument to justify each of your answers.
Which of the properties of anonymity, neutrality, and monotonicity are satisfied by the Borda count? Which of these three properties are not satisfied? Give a convincing argument to justify each of your answers.
Do either of your answers to Questions 1 or 2 contradict Theorem 7.1.12? Explain.
In this section, we have seen that things become more complicated when we consider more than two candidates. In an election with only two candidates, a vote for one implicitly ranks the other candidate second. With more than two candidates, not only can societal preferences be more diffuse, but they are also more complex than can be captured with simple plurality voting. It is possible for a candidate who is the least desirable choice of an overwhelming majority of the voters to win if there are enough other candidates.
In the next section, we will explore additional voting systems and fairness criteria that attempt to overcome the shortcomings of the Borda count.
