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
© 2009, W.H. Freeman and Company
Mathematical Literacy in
Today’s World, 8th ed.
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.
Chapter 9: Social Choice: The Impossible Dream
Majority Rule and Condorcet’s Method
 Majority Rule
 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.
May’s Theorem – Among all
 It is monotone.
two-candidate voting
Monotone means that if a new
systems that never result
election were held and a single voter
in a tie, majority rule is
were to change his or her ballot from
the only one that treats
voting for the losing candidate to
all voters and both
voting for the winning candidate (and
candidates equally and is
everyone else voted the same), the
monotone.
outcome would be the same.
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.
Chapter 9: Social Choice: The Impossible Dream
Majority Rule and Condorcet’s Method
 Condorcet’s Method
 This method requires that each candidate go head-to-head with
each of the other candidates. Condorcet’s Voting Paradox – With
 For the two candidates in each
three or more candidates, there
contest, record who would win
are elections in which
(using majority rule) from each
Condorcet’s method yields no
ballot cast. To satisfy
winners.
Condorcet, the winning
candidate must defeat every
A beats B, 2 out of 3; and
other candidate one-on-one.
B beats C, 2 out of 3; and
 The Marquis de Condorcet
C beats A, 2 out of 3 — No winner!
(1743 – 1794) was the first to
Rank
Number of Voters (3)
realize the voting paradox:
If A is better than B, and B is
First
A
B
C
better than C, then A must be
Second
B
C
A
better than C. Sometimes C is
Third
C
A
B
better than A—not logical!
 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?
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.
Chapter 9: Social Choice: The Impossible Dream
Other Voting Systems for Three or More Candidates
 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:
1.
2.
3.
4.
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
Chapter 9: Social Choice: The Impossible Dream
Plurality Voting and the Condorcet Winning Criterion
 Plurality Voting
 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
B
Second
B
B
C
A
Third
C
A
A
C
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 = 4
A wins.
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.
 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.
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:
1. If there is no Condorcet winner (often the case) - or 2. 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.
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=2+2+2+0+0=6
B=1+1+1+2+2=7
C=0+0+0+1+1=2
B has the most, B wins.
Rank
Number of Voters (5)
Points
First
A
A
A
B
B
2
Second
B
B
B
C
C
1
Third
C
C
C
A
A
0
Another way: Count the occurrences of letters below the
candidate—for example, there are 7 “boxes” below B
Example:
Who is the winner using Borda count?
The winner is A.
The sum is 36 + 25 + 20 = 81. This is the same as the
product of 3 (2 + 1 + 0) and 27 (number of voters).
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
Original Borda Score: A=6, B=5, C=4 Suppose the last
Rank
Number of Voters (5)
First
A
A
A
C
C
Second
B
B
B
B
B
Third
C
C
C
A
A
two voters change
their ballots
(reverse the order
of just the losers).
This should not
change the winner.
and did not switch with A!
New Borda Score: A= 6, B=7, C=2
Rank
Number of Voters (5)
First
A
A
A
B
B
Second
B
B
B
C
C
Third
C
C
C
A
A
Chapter 9: Social Choice: The Impossible Dream
Sequential Pairwise Voting and the Pareto Condition
 Sequential Pairwise Voting
 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?
Rank
Number of Voters (3)
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).
First
A
C
B
Second
B
A
D
A vs. B
A vs. C
Third
D
B
C
II
I
Fourth
C
D
A
I
A wins; B is
deleted.
II
C wins; A is
deleted.
C vs. D
Candidate D
I
II
wins for this
D wins; C is agenda.
deleted.
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?
 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.
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
B
A
D
Third
D
B
C
Fourth
C
D
A
Different agenda orders can
change the outcomes. For
example, agenda D, C, B, A
results in A as the winner.
Chapter 9: Social Choice: The Impossible Dream
The Hare System and Monotonicity
 The Hare System
 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.
Rank
5
4
3
1
For the Hare system, delete
the candidate with the least
first-place votes:
First
A
C
B
B
A = 5, B = 4, and C = 4
Second
B
B
C
A
Third
C
A
A
C
Since B and C are tied for the
least first place votes, they are
both deleted and A wins.
Number of Voters (13)
Example:
Who is the winner using the Hare system?
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.
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.
Example:
Who is the winner using the plurality runoff?
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.
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
A
Second
B
B
C
B
Third
C
A
A
C
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.”
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
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.
 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.
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 (1 + 2 + 3 + 4). Who is
the winner and how would they be ranked?
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).
Solution to b):
A has 1+ 4 = 5 approval votes. B has 2 + 4 = 6
approval votes. C has 4 approval votes. D has1+ 2 + 3 =
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).