Suppose Skip, Norm, and Jesse are all running for President of the 10,000 Lakes Club, with the preferences of the 100 members of the club as shown in Table 7.3.2.
Table7.3.2.The preference schedule schedule for the 10,000 Lakes Club presidency.
Rank
35
28
20
17
1
N
S
J
J
2
S
N
N
S
3
J
J
S
N
What would be the outcome of the election under majority rule?
What would be the outcome of the election under plurality?
What would be the outcome of the election under the Borda count?
Which candidate is ranked first by the largest number of voters?
Which candidate is ranked last by the largest number of voters?
In a head-to-head contest between just Skip and Norm, who would win?
In a head-to-head contest between just Skip and Jesse, who would win?
In a head-to-head contest between just Norm and Jesse, who would win?
Does anything about your answers to Questions 1-8 strike you as being strange or unusual?
In Activity 7.3.1, we saw that the plurality method can fail to elect a candidate who would win a head-to-head matchup against all other candidates. Perhaps worse, we also saw that plurality can elect a candidate who would lose a head-to-head matchup against all other candidates. This has struck voting theorists as unfair, and we make the following definition, named after Marie Jean Antoine Nicolas de Caritat, the Marquis de Condorcet.
Definition7.3.3.
A Condorcet winner is a candidate in an election who would win a head-to-head contest (with the winner decided by majority rule) against each of the other candidates.
A Condorcet loser is a candidate in an election who would lose a head-to-head contest (with the winner decided by majority rule) against each of the other candidates.
A voting system that will always elect a Condorcet winner, whenever one exists, is said to satisfy the Condorcet winner criterion (CWC).
A voting system that will never elect a Condorcet loser is said to satisfy the Condorcet loser criterion (CLC).
In a head-to-head contest between just candidates A and B, who would win?
In a head-to-head contest between just candidates B and C, who would win?
In a head-to-head contest between just candidates A and C, who would win?
Does anything about Questions 1-3 strike you as unusual?
Is there a Condorcet winner and/or loser in this election? Explain.
Investigation7.3.6.
Explain why a candidate who received a majority of first-place votes is also a Condorcet winner.
Does your answer to Question 1 imply that majority rule satisfies the CWC? If so, explain why. Otherwise, give an example to show that majority rule can violate the CWC.
Does your answer to Question 1 imply that majority rule satisfies the CLC? If so, explain why. Otherwise, give an example to show that majority rule can violate the CLC.
Are there special types of elections for which majority rule does satisfy the CWC? Give a convincing argument to justify your answer.
Use your answer to Question 1 to explain why any voting system that violates the majority criterion ( Definition 7.2.15) must also violate the CWC.
Use your answer to Question 5 to explain why the Borda count violates the CWC.
In order to find a voting system that satisfies the CWC and CLC, let’s return to the Dunder Mifflin presidential election in Exploration 7.2.9.
Exploration7.3.7.
Consider the following proposed voting system, using Table 7.2.10 as the preference schedule.
Step 1: List all possible head-to-head matchups between candidates.
Step 2: Determine the winner of each head-to-head matchup. Award 1 point for each win, 0 points for a loss, and 1/2 point to each in the event of a tie.
Step 3: The candidate with the most points from Step 2 wins.
Consider the method described above.
Under this method, who wins the presidency of Dunder Mifflin?
Under this method, what societal preference order is produced?
Is there a Condorcet winner and/or loser in this election? Explain.
The method described above is known as the method of pairwise comparisions (PWC). What strikes you as being different or unusual about it, especially compared to the plurality and Borda count systems we’ve already explored?
Investigation7.3.8.
Could a Condorcet winner ever lose a head-to-head contest with another candidate? Why or why not?
What does your answer to Question 1 allow you to conclude about the method of pairwise comparisons and the CWC?
Does the method of pairwise comparisons satisfy the CLC? If so, explain why. Otherwise, give an example of a preference schedule for the method of pairwise comparisons could elect a Condorcet loser.
As we have just seen, the method of pairwise comparisons satisfies the CWC. This means that it will always elect a Condorcet winner if one exists. However, when a Condorcet winner does not exist, strange things can happen.
Investigation7.3.9.
Consider the preference schedule given in Table 7.3.10.
Table7.3.10.A preference schedule.
Rank
10
3
7
6
1
C
E
D
B
2
E
A
A
A
3
D
D
C
C
4
A
C
B
D
5
B
B
E
E
Who wins this election using the method of pairwise comparisons?
Is there a Condorcet winner? A Condorcet loser?
Due to a scandal following the counting of the votes, candidate D withdraws. The election is re-run using the votes already cast, again with the method of pairwise comparisons. Who wins?
Does this seem strange?
The phenomenon observed in Investigation 7.3.9 is an example of a violation of a new fairness criterion, the independence of irrelevant alternatives. It’s more subtle than some of the others, so we’ll present a few versions of it.
Definition7.3.11.
If a voting system has the property that the societal preference between any two candidates depends only on the voters’ preferences between those two candidates, then the system is said to satisfy the independence of irrelevant alternatives criterion (IIA).
Put another way, a voting system satisfies IIA if some or all of the voters in an election change their preference ballots but no voter changes their preference between two candidates \(A\) and \(B\text{,}\) then the societal preference between \(A\) and \(B\) must also remain unchanged.
Or, if an election is run and produces \(X\) as the winner, and later a non-winning candidate \(Y\) drops out, then \(X\) should still be the winner of the election.
IIA is often interpreted as saying that if a candidate (\(A\)) would win an election, and a new candidate (\(B\)) were added to the ballot, then either \(A\) or \(B\) should win the election. A further delightful illustration of a violation of IIA is attributed to Sidney Morgenbesser 2
en.wikipedia.org/wiki/Sidney_Morgenbesser
:
After finishing dinner, Sidney Morgenbesser decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie at which point Morgenbesser says “In that case I’ll have the blueberry pie.”
Let’s consider our current systems and IIA.
Investigation7.3.12.
Does plurality satisfy IIA? Why or why not?
Does the Borda count satisfy IIA? Why or why not?
Does PWC satisfy IIA? Why or why not?
Exploration7.3.13.
Is the method of pairwise comparisons anonymous? Monotone? Does it satisfy the majority criterion? Justify your answers.
Let’s take stock of where we’re at.
Activity7.3.14.
Fill in Table 7.3.15, indicating whether each voting system satisfies the given criteria.
Table7.3.15.Voting systems and fairness criteria
Anonymous
Neutral
Monotone
Majority
CWC
CLC
IIA
Majority rule
Plurality
Borda count
PWC
Subsection7.3.1.2Instant Runoff Voting and Arrow’s Theorem
In this section, we’ll examine one last (new) voting system, called instant runoff voting (IRV) or ranked choice voting (RCV). This system was proposed in the mid-1800s by Thomas Hare, and has slowly grown in popularity. Other than plurality, it is likely the most widely used system for choosing elected officials in the U.S. For example, the City of Minneapolis 3
The instant runoff voting (IRV) system works as follows.
Each voter in the election submits their entire preference order.
If a candidate has a majority of first-place votes, they are declared the winner.
If no candidate has a majority of first-place votes, then the candidate (or candidates, in the case of a tie) with the fewest first-place votes is eliminated from each voter’s preference order, and the remaining candidates are moved up, yielding a new preference schedule.
Step 2 is repeated until a single candidate remains. That candidate is declared the winner.
Activity7.3.17.
Consider the hypothetical preference schedule shown in Table 7.3.18.
Table7.3.18.A hypothetical election
Rank
6
7
9
3
1
A
B
C
A
2
B
C
B
C
3
C
A
A
B
Under IRV, which candidate is eliminated first?
Under IRV, which candidate is eliminated second?
Who would win the election under IRV? What would be the resulting societal preference order?
As has been our wont, let’s explore which fairness criteria are satisfied by IRV.
Investigation7.3.19.
Use Definition 7.2.17 to write a thorough explanation of why IRV is both neutral and anonymous.
Investigation7.3.20.
Explain why IRV satisfies the majority criterion.
Activity7.3.21.
Consider the election run in Activity 7.3.17. Suppose that the three voters in the rightmost column of Table 7.3.18 got wind of candidate \(C\)’s support and switched their votes to \(C \succ A \succ B\) to be on the side of the winner.
With these new preferences, who would win the election under IRV?
Compare your answer to Question 1 of this activity to Question 3 of Activity 7.3.17. What conclusions can you draw?
about this phenomenon. How do they respond to a potential critique?
The phenomonon observed in Activity 7.3.21 suggests our last fairness criterion: the monotonicity criterion.
Definition7.3.22.
A voting system is said to satisfy the monotonicity criterion if when a candidate \(X\) wins, and on a recount or reelection, only changes favorable to \(X\) occur, then \(X\) will still win. That is, changes that only favor the winner should not change the winner.
Investigation7.3.23.
Consider an election between three candidates with the preference schedule shown in Table 7.3.24.
Table7.3.24.A hypothetical election.
Rank
1
2
2
1
A
B
C
2
B
A
A
3
C
C
B
Is there a Condorcet winner in this election?
Who would win the election under IRV?
Does IRV satisfy the Condorcet winner criterion? Use your answers to Questions 1 and 2, together with Definition 7.3.3 (and what we know about when implications are false! Recall Definition 3.1.11.).
Investigation7.3.25.
What about Dunder Mifflin? Who wins the company presidency with the preference schedule in Table 7.2.10 under IRV?
Given all the investigations we’ve done into the Dunder Mifflin presidency, write an argument to the company’s board of directors arguing for a particular voting system to be used to choose the company’s president. Your argument should make some allusion to the fairness criteria we’ve explored.
At this point, you may be wondering which voting system is best. We saw a decisive answer if our election has only two candidates (the familiar majority rule), but when the election has more than two candidates, things have gotten complicated. Plurality, majority rule, the Borda count, PWC, and IRV all fail to satisfy at least one of our criteria.
proved the following landmark result. It states that our quest is hopeless! There is no fairest voting system. Each has tradeoffs which make it suitable in certain circumstances and unsuitable in others.
Arrow’s Impossibility Theorem.
In an election with more than two candidates, it is impossible for a voting system to satisfy monotonicity, neutrality, anonymity, IIA, the majority criterion, and the CWC.
In this section, we explored additional voting systems (the method of pairwise comparisions and instant runoff voting) and fairness criteria (IIA and the monotonicity criterion). As we saw that none of our systems satisfied all of our criteria, we were faced with a question: is there any voting system out there that is completely fair? Arrow’s landmark theorem tells us that no democratic voting system can satisfy all of our criteria. This does not mean that our explorations have been fruitless —instead, we’ll need to consider which criteria are most important to us when selecting a voting system, and choose accordingly.
ExercisesExercises
1.
Consider the hypothetical election below.
Table7.3.26.
Rank
8
6
4
2
1
A
C
D
B
2
B
A
C
C
3
D
B
B
A
4
C
D
A
D
Who wins the election using the method of pairwise comparisons?
Suppose that after a scandal emerges, candidate \(D\) drops out and the results are re-tabulated. Who wins the election?
Does the answer to the previous question give an example of a violation of one of our fairness criteria? Explain.
Suppose instead that, before the original election is run, the four voters in the third column switch their preference ballot to \(C \succ D \succ B \succ A\text{.}\) Who wins the election? Is this an example of a violation of one of our fairness criteria? Explain.
Using the original preference schedule, who wins the election using instant runoff voting? Explain.
2.
One criticism of non-plurality voting systems is that they are too complex, or ballots too difficult to understand. Find an article that makes this point and respond to it in a well-reasoned paragraph or two. Do you agree? Disagree?
