1. Chapter 9: Social Choice: The Impossible Dream Lesson Plan
For All Practical Purposes
Voting and Social Choice
Majority Rule and Condorcet’s Method
Other Voting Systems for Three or More Candidates
Plurality Voting
Borda Count
Sequential Pairwise Voting
Hare System
Insurmountable Difficulties: Arrow’s Impossibility Theorem
A Better Approach? Approval Voting
Mathematical Literacy in Today’s World, 8th ed. 1
© 2009, W.H. Freeman and Company

2. Chapter 9: Social Choice: The Impossible Dream Voting and Social Choice
Social Choice Theory
Social choice deals with how groups can best arrive at decisions.
The problem with social choice is finding good procedures that will turn individual preferences for different candidates into a single choice by the whole group.
Example: Selecting a winner of an election using a good procedure that will result in an outcome that “reflects the will of the people”
Preference List Ballot
A preference list ballot consists of a rank ordering of candidates showing the preferences of one of the individuals who is voting.
A vertical list is used with the most preferred candidate on top and the least preferred on the bottom.
Throughout the chapter, we assume the number of voters is odd (to help simplify and focus on the theory). Furthermore, in the real world, the number of voters is often so large that ties seldom occur. 2

4. Chapter 9: Social Choice: The Impossible Dream Majority Rule and Condorcet’s Method
Majority rule for elections with only two candidates (and an odd number of voters) is a voting system in which the candidate preferred by more than half the voters is the winner.
Three Desirable Properties of Majority Rule
All voters are treated equally.
Both candidates are treated equally.
It is monotone.
May’s Theorem – Among all two-candidate voting systems that never result in a tie, majority rule is the only one that treats all voters and both candidates equally and is monotone.
Monotone means that if a new election were held and a single voter were to change his or her ballot from voting for the losing candidate to voting for the winning candidate (and everyone else voted the same), the outcome would be the same. 4

5. Voting Criteria Definitions/rules/theorems that appear in the chapter are as follows.
• In a dictatorship, all ballots except that of the dictator are ignored.
• In imposed rule, candidate X wins regardless of who votes for whom.
• In minority rule, the candidate with the fewest votes wins.
• When there are only two candidates or alternatives, May’s theorem states that majority rule is the only voting method that satisfies three desirable properties, given an odd number of voters and no ties.

6. Chapter 9: Social Choice: The Impossible Dream Majority Rule and Condorcet’s Method
This method requires that each candidate go head-to-head with each of the other candidates.
Condorcet’s Voting Paradox – With three or more candidates, there are elections in which Condorcet’s method yields no winners.
A beats B, 2 out of 3; and
B beats C, 2 out of 3; and
C beats A, 2 out of 3 — No winner!
Rank Number of Voters (3)
First A B C
Second B C A
Third C A B
For the two candidates in each contest, record who would win (using majority rule) from each ballot cast. To satisfy Condorcet, the winning candidate must defeat every other candidate one-on-one.
The Marquis de Condorcet (1743 – 1794) was the first to realize the voting paradox: If A is better than B, and B is better than C, then A must be better than C. Sometimes C is better than A—not logical! 6

7. In Condorcet’s method, a candidate is declared the winner if he or she can defeat every other candidate in a one-on-one competition using majority rule. That is, the winner of each one-on-one competition will have over 50% of the votes.
Example:
Determine if there is a winner using Condorcet’s method. If so, who is it?

8. Solution: You must determine the outcome of three one-on-one competitions. The candidates not considered in each one-on-one competition can be ignored.
Since A can defeat both B and C in a one-on-one competition, A is the winner by the Condorcet method.

9. Chapter 9: Social Choice: The Impossible Dream Other Voting Systems for Three or More Candidates
When there are three or more candidates, it is more unlikely to have a candidate win with a majority vote.
Many other voting methods exist, consisting of reasonable ways to choose a winner; however, they all have shortcomings.
We will examine four more popular voting systems for three or more candidates:
Four voting systems, along with their shortcomings:
Plurality Voting and the Condorcet Winning Criterion
The Borda Count and Independence of Irrelevant Alternatives
Sequential Pairwise Voting and the Pareto Condition
The Hare System and Monotonicity 9

10. Chapter 9: Social Choice: The Impossible Dream Plurality Voting and the Condorcet Winning Criterion
Only first-place votes are considered.
Even if a preference list ballot is submitted, only the voters’ first choice will be counted—it could have just been a single vote cast.
The candidate with the most votes wins.
The winner does not need a majority of votes, but simply have more votes than the other candidates.
Example: Find the plurality vote of the 3 candidates and 13 voters.
Number of Voters (13)
Rank 5 4 3 1
First A C B
Second
Third
The candidate with the most first-place votes wins. Count each candidate’s first-place votes only. (A has the most.)
A = 5, B = 4, C = A wins. 10

11. In plurality voting, the candidate with the most first-place votes on the preference list ballots is the winner. We do not take into account the voters’ preferences for the second, third, etc., places.

12. Solution:
A has 9 first-place votes. B has 8 first-place votes. C has 10 first-place votes. Since C has the most first-place votes, C is the winner.

13. Chapter 9: Social Choice: The Impossible Dream Plurality Voting and the Condorcet Winning Criterion
Example: 2000 Presidential Election (Plurality fails CWC.)
Condorcet Winner Criterion (CWC) is satisfied if either is true:
If there is no Condorcet winner (often the case) - or -
If the winner of the election is also the Condorcet winner
This election came down to which of Bush or Gore would carry Florida. Result: George W. Bush won by a few hundred votes.
Gore, however, was considered the Condorcet winner:
It is assumed if Al Gore was pitted against any one of the other three candidates, (Bush, Buchanan, Nader), Gore would have won.
Manipulability occurs when voters
misrepresents their preference
rather than “throw away” their vote. 13

14. Chapter 9: Social Choice: The Impossible Dream Borda Count and Independence of Irrelevant Alternatives
The Borda Count
Borda Count is a rank method of voting that assigns points in a nonincreasing manner to the ordered candidates on each voter’s preference list ballot and then add these points to arrive at a group’s final ranking.
For n candidates, assign points as follows:
First-place vote is worth n − 1 points, second-place vote is worth n − 2 points, and so on down to…Last place vote is worth n − n = 0, zero points.
The candidate’s total points are referred to as his/her Borda score.
Example: Total the Borda score of each candidate.
A = = B = = C = = 2
B has the most, B wins.
Rank
Number of Voters (5)
Points
First A B 2
Second C 1
Third
Another way: Count the occurrences of letters below the candidate—for example, there are 7 “boxes” below B 14

15. Example: Who is the winner using Borda count?

16. The winner is A.
The sum is = 81. This is the same as the product of 3 ( ) and 27 (number of voters).

17. Chapter 9: Social Choice: The Impossible Dream Borda Count and Independence of Irrelevant Alternatives
Independence of Irrelevant Alternatives (Borda fails IIA.)
A voting system is said to satisfy independence of irrelevant alternatives (IIA) if it is impossible for candidate B to move from nonwinner status to winner status unless at least one voter reverses the order in which he or she had B and the winning candidate ranked.
If B was a loser, B should never become a winner, unless he moves ahead of the winner (reverses order) in a voter’s preference list.
Example showing that Borda count fails to satisfy IIA:
B went from loser to winner and did not switch with A!
Suppose the last two voters change their ballots (reverse the order of just the losers). This should not change the winner.
Original Borda Score: A=6, B=5, C=4
New Borda Score: A= 6, B=7, C=2
Rank
Number of Voters (5)
First A C
Second B
Third
Rank
Number of Voters (5)
First A B
Second C
Third 17

18. Chapter 9: Social Choice: The Impossible Dream Sequential Pairwise Voting and the Pareto Condition
Sequential pairwise voting starts with an agenda and pits the first candidate against the second in a one-on-one contest.
The losers are deleted and the winner then moves on to confront the third candidate in the list, one on one.
This process continues throughout the entire agenda, and the one remaining at the end wins.
Example: Who would be the winner using the agenda A, B, C, D for the following preference list ballots of three voters?
Using the agenda A, B, C, D, start with A vs. B and record (with tally marks) who is preferred for each ballot list (column).
Rank
Number of Voters (3)
First A C B
Second D
Third
Fourth
A vs. B
II I
A vs. C
I II
C vs. D
I II
Candidate D wins for this agenda.
A wins; B is deleted.
C wins; A is deleted.
D wins; C is deleted. 18

19. An agenda is the listing (in some order) of the candidates
An agenda is the listing (in some order) of the candidates. Sequential pairwise voting pits the first candidate against the second in a one-on-one contest. The winner goes on to confront the third candidate on the agenda, while the loser is eliminated. The candidate remaining at the end is the winner. The choice of the agenda can affect the result.
Example:
Who is the winner using sequential pairwise voting with the agenda C, A, B?

20. Solution
In sequential pairwise voting with the agenda C, A, B, we first pit C against A. There are 10 voters who prefer C to A and 17 prefer A to C. Thus, A wins by a score of 17 to 10. C is therefore eliminated, and A moves on to confront B. There are 19 voters who prefer A to B and 8 prefer B to A. Thus, A wins by a score of 19 to 8. Thus, A is the winner by sequential pairwise voting with the agenda C, A, B.

21. Chapter 9: Social Choice: The Impossible Dream Sequential Pairwise Voting and the Pareto Condition
Pareto Condition (Sequential Pairwise fails Pareto.)
Pareto condition states that if everyone prefers one candidate (in this case, B) to another candidate (D), then this latter candidate (D) should not be among the winners of the election.
Pareto condition is named after Vilfredo Pareto (1848 – 1923), Italian economist.
From the last example:
D was the winner for the agenda A, B, C, D.
However, each voter (each of the three preference lists columns) preferred B over D.
If everyone preferred B to D, then D should not have been the winner! Not fair!
Rank
Number of Voters (3)
First A C B
Second D
Third
Fourth
Different agenda orders can change the outcomes. For example, agenda D, C, B, A results in A as the winner. 21

22. Chapter 9: Social Choice: The Impossible Dream The Hare System and Monotonicity
The Hare system proceeds to arrive at a winner by repeatedly deleting candidates that are “least preferred” (meaning at the top of the fewest ballots).
If a single candidate remains after all others have been eliminated, he/she alone is the winner.
If two or more candidates remain and they all would be eliminated in the next round, then these candidates would tie.
For the Hare system, delete the candidate with the least first-place votes:
A = 5, B = 4, and C = 4
Since B and C are tied for the least first place votes, they are both deleted and A wins.
Number of Voters (13)
Rank 5 4 3 1
First A C B
Second
Third 22

23. Example: Who is the winner using the Hare system?

24. Solution A has 9 first-place votes. B has 8 first-place votes
Solution A has 9 first-place votes. B has 8 first-place votes. C has 10 first-place votes. Since B has the least number of first-place votes, B is eliminated. Candidates A and C move up as indicated to form a new table.
A now has 17 first-place votes. C now has 10 first-place votes. Thus, A is the winner by the Hare system.

25. What happens if there is a tie for second place using the Hare system?
Plurality runoff is the voting system in which there is a runoff between the two candidates receiving the most first-place votes. In the case of ties between first or second, three candidates participate in the runoff. This system is basically a one-step process. Determine the candidates that are in the runoff, create new preference list ballots by deleting the candidates not in the runoff, and then determine which candidate has the plurality of the votes.

26. Example: Who is the winner using the plurality runoff?

27. Solution: A has 9 first-place votes. B has 8 first-place votes
Solution: A has 9 first-place votes. B has 8 first-place votes. C has 10 first-place votes. Since A and C have the highest first- and second-place votes, B is eliminated. Candidates A and C move up as indicated to form a new table.
A now has 17 first-place votes. C now has 10 first-place votes. Thus, A is the winner by the plurality runoff.

28. Chapter 9: Social Choice: The Impossible Dream The Hare System and Monotonicity
Monotonicity (The Hare system fails monotonicity.)
Monotonicity says that if a candidate is a winner and a new election is held in which the only ballot change made is for some voter to move the former winning candidate higher on his or her ballot, then the original winner should remain a winner.
In a new election, if a voter moves a winner higher up on his preference list, the outcome should still have the same winner.
Number of Voters (13)
Rank 5 4 3 1
First A C B
Second
Third
In the previous example, A won. For the last voter, move A up higher on the list (A and B switch places). Round 1: B is deleted. Round 2: C moves up to replace B on the third column. However, C wins—this is a glaring defect!
The Hare system, introduced by Thomas Hare in 1861, was known by names such as the “single transferable vote system.” In 1962, John Stuart Mill described the Hare system as being “among the greatest improvements yet made in the theory and practice of government.” 28

29. Chapter 9: Social Choice: The Impossible Dream Insurmountable Difficulties: Arrow’s Impossibility Theorem
Arrow’s Impossibility Theorem
Kenneth Arrow, an economist in 1951, proved that finding an absolutely fair and decisive voting system is impossible.
With three or more candidates and any number of voters, there does not exist—and there never will exist—a voting system that always produces a winner, satisfies the Pareto condition and independence of irrelevant alternatives (IIA), and is not a dictatorship.
If you had an odd number of voters, there does not exist—and there never will exist—a voting system that satisfies both the CWC and IIA and that always produces at least one winner in every election.
Kenneth Arrow 29

30. Chapter 9: Social Choice: The Impossible Dream A Better Approach
Chapter 9: Social Choice: The Impossible Dream A Better Approach? Approval Voting
Approval Voting
Under approval voting, each voter is allowed to give one vote to as many of the candidates as he or she finds acceptable.
No limit is set on the number of candidates for whom an individual can vote; however, preferences cannot be expressed.
Voters show disapproval of other candidates simply by not voting for them.
The winner under approval voting is the candidate who receives the largest number of approval votes.
This approach is also appropriate in situations where more than one candidate can win, for example, in electing new members to an exclusive society such as the National Academy of Sciences or the Baseball Hall of Fame. Approval voting is also used to elect the secretary general of the United Nations.
Approval voting was proposed independently by several analysts in 1970s. 30

31. In approval voting, each voter may vote for as many candidates as he or she chooses. The candidate with the highest number of approval votes wins the election. A candidate receives an X if he or she has obtained approval from one or more of the voters. There are two types of tables presented in this section. One is where the voter has an identification number (1, 2, 3, …), and the other indicates how many voters voted for the same combination of nominees.

32. Example: Suppose the following table is the result of approving four nominees. Treat the table in two ways.
a) Assume there are 4 voters. Who is the winner and how would they be ranked?
b) Assume there are 10 voters ( ). Who is the winner and how would they be ranked?

33. Solution to a):
A has 2 approval votes. B has 2 approval votes. C has 1 approval vote. D has 3 approval votes. Since D has the most approval votes, D is the winner. Ranking the candidates we have, D (3), A and B (2), and C (1).

34. Solution to b):
A has 1+ 4 = 5 approval votes. B has = 6 approval votes. C has 4 approval votes. D has = 6 approval votes. Since B and D have the most approval votes, B and D tie as winners. Ranking the candidates we have, B and D (6), A (5), and C (4).