EXCURSIONS IN MODERN
MATHEMATICS
SIXTH EDITION
Peter Tannenbaum
1
CHAPTER 1
THE MATHEMATICS OF
VOTING
2
The Paradoxes of Democracy
The Mathematics of Voting
Outline/learning Objectives
3





Construct and interpret a preference schedule for an
election involving preference ballots.
Implement the plurality, Borda count, plurality-withelimination, and pairwise comparisons vote counting
methods.
Rank candidates using recursive and extended
methods.
Identify fairness criteria as they pertain to voting
methods.
Understand the significance of Arrows’ impossibility
theorem.
4
THE MATHEMATICS OF
VOTING
1.1 Preference Ballots and Preference Schedules
Voting Theory
5
Why Vote?
 Think about all the elections you can vote in:
 Presidential
 Local
elections
 School board
 Homecoming queen
 American Idol

Clearly, not all voting is equal
Voting Theory
6


Casting a vote is only part of the story. What
matters more is how the votes are counted to
determine a winner.
To analyze the various voting methods we need:
 Candidates
– the choices
 Voters – the ones who are voting
 Ballots – the way the votes are collected
Voting Theory
7

In 1940, Kenneth Arrow discovered an incredible
fact: for elections involving three or more
candidates, there is no consistently fair democratic
method for choosing a winner.


In fact, a method for determining election results that is
democratic and always fair is a mathematical impossibility.
This is known as Arrow’s Impossibility Theorem.
Preference Ballots and Schedules
8


Preference ballots
A ballot in which the voters are asked to rank the
candidates in order of preference.
Linear ballot
A ballot in which ties are not allowed.
Preference Ballots and Schedules
9


An example of a preference ballot is each person’s
vote in our math class party election.
Preference ballots allow voters to express an
opinion on all candidates instead of just choosing
their choice.
Ballot
1st
2nd
3rd
4th
Preference Ballots and Schedules
10
Ballot
Ballot
Ballot
Ballot
Ballot
Ballot
Ballot
Ballot
Ballot
C
A
C
B
C
C
A
D
A
B
B
D
D
D
B
B
B
B
D
C
B
A
B
D
C
C
C
A
D
A
C
A
A
D
A
D
Ballot
Ballot
Ballot
Ballot
Ballot
Ballot
Ballot
Ballot
Ballot
D
C
A
A
C
C
D
B
C
B
D
B
B
B
B
B
D
B
C
B
C
C
D
D
C
A
D
A
A
D
D
A
A
A
C
A
18 voters
Preference Ballots and Schedules
11

There are a limited number of ways the candidates
can be ranked, so some ballots may repeat. If we
collect all the repeats and organize them in a table,
we get a preference schedule.
Number
of Voters
1st
2nd
3rd
4th
Preference Ballots and Schedules
12
Number
of Voters
4
5
4
2
3
1st
C
A
C
B
D
2nd
B
B
D
D
B
3rd
D
C
B
A
C
4th
A
D
A
C
A
Preference Ballots and Schedules
13
Preference Ballots and Schedules
14

A preference schedule:
Preference Ballots and Schedules
15
Things to keep in mind:
Voter preferences are transitive


If I like Sprite better than Dr. Pepper and I
like Dr. Pepper more than Mt. Dew, then I
like Sprite better than Mt. Dew.
If we need to know which candidate a
voter would vote for if it all came down to
candidate X and candidate Y, we just look
at where X and Y are on that person’s
ballot.
Preference Ballots and Schedules
16
Things to keep in mind:
Relative preferences are not affected by elimination
of a candidate.
 If
a candidate drops out of a race, then the votes shift
up accordingly.
 Example: If I remove Starburst from the candy options
(because the wrappers will end up everywhere), to
recalculate a winner, I just move everyone’s votes up
accordingly.
Preference Ballots and Schedules
17
Relative
Preferences by
elimination of
one or more
candidates
Preference Ballots and Schedules
18


How many people voted in this election?
If D gets sick and can’t run, what is the new
schedule?
51
48
5
1st Choice
A
D
E
2nd Choice
B
C
C
3rd Choice
C
B
D
4th Choice
D
A
B
5th Choice
E
E
A
Preference Ballots and Schedules
19

The Mathematics Society is holding an election for the
president. The three candidates are A, B, and C. Forty-five
percent of voters like A the most and B the least. Thirty percent
of voters like B the most and C the least. Twenty-five percent
of voters like C the most and A the least. Write out the
preference schedule for this election.
1st Choice
2nd Choice
3rd Choice
Preference Ballots and Schedules
20

In an election involving 6 candidates, what is the
maximum number of columns possible in the
preference schedule?
N!
21
THE MATHEMATICS OF
VOTING
1.2 The Plurality Method
The Plurality Method
22


Plurality method
In the plurality method, all we care about is firstplace votes. The candidate with the most first-place
votes wins.
Plurality candidate
The Candidate with the most 1st place votes
The Plurality Method
23

The vast majority of our elections are decided using
the plurality method. Since the only votes that count
are first-place votes, we don’t bother to rank the
other candidates.
The Plurality Method
24

Find the winner of this election using the plurality
method:
Number
of Voters
4
5
4
2
3
1st
C
A
C
B
D
2nd
B
B
D
D
B
3rd
D
C
B
A
C
4th
A
D
A
C
A
The Plurality Method
25


Majority rule
The candidate with a more than half the votes
should be the winner.
Majority candidate
The candidate with the majority of 1st place votes .
The Plurality Method
26

The plurality method is appealing because it is
simple and it is a natural extension of the principle
of majority rule.


In a democratic election between 2 candidates, the candidate
with a majority of the votes should be the winner. The
candidate with a majority of first-place votes is called the
majority candidate.
However, with 3 or more candidates, there is no
guarantee that one candidate will win a majority of
votes.
The Plurality Method
27
The Majority Criterion (a fairness criterion)
If candidate X has a majority of the 1st place votes,
then candidate X should be the winner of the
election.
Good News: The plurality method satisfies the majority
criterion!
Bad News: The plurality method fails a different
fairness criterion.
The Plurality Method
28


Under the plurality method, a majority candidate is
guaranteed to be the winner.
Why?
The Plurality Method
29

There are widely used voting methods that can
produce violations of the majority criterion.
Specifically, a violation of the majority criterion
occurs in an election in which there is a majority
candidate but that candidate does not win the
election. If this can happen under some voting
method, then the voting method itself violates the
majority criterion.

Note: violations may occur, not that they always will occur.
The Plurality Method
30

Other than the majority criterion, the plurality
method has little appeal and is undesirable when
choosing between more than two candidates.
The Plurality Method
31

The principal weakness of the plurality method is
that it fails to take into consideration a voter’s other
preferences beyond first choice and in doing so can
lead to some very bad election results.
The Plurality Method
32

Tasmania State University has a superb marching
band. They are so good that they have invitations to
perform at five different bowl games: the Rose Bowl
(R), the Hula Bowl (H), the Fiesta Bowl (F), the Orange
Bowl (O), and the Sugar Bowl (S). An election is held
among the 100 members of the band to decide in
which bowl game they will perform.
The Plurality Method
33

Tasmania State University has a superb marching band. They
are so good that they have invitations to perform at five
different bowl games: the Rose Bowl (R), the Hula Bowl (H), the
Fiesta Bowl (F), the Orange Bowl (O), and the Sugar Bowl (S).
An election is held among the 100 members of the band to
decide in which bowl game they will perform.
# of voters
49
48
3
1st choice
R
H
F
2nd choice
H
S
H
3rd choice
F
O
S
4th choice
O
F
O
5th choice
S
R
R
The Plurality Method
34

Find the winner using the plurality method.

Is this a good outcome? Why or why not?
# of voters
49
48
3
1st choice
R
H
F
2nd choice
H
S
H
3rd choice
F
O
S
4th choice
O
F
O
5th choice
S
R
R
The Plurality Method
35


By contrast, the Hula Bowl has 48 first-place votes
and 52 second-place votes. Common sense tells us
that the Hula Bowl is a far better choice to
represent the wishes of the entire band.
In fact, if we compare the Hula Bowl with any other
bowl on a head-to-head basis, the Hula Bowl is
always the preferred choice.
The Plurality Method
36

For example, compare the Hula Bowl to the Rose
Bowl.
# of voters
49
48
3
1st choice
R
H
F
2nd choice
H
S
H
3rd choice
F
O
S
4th choice
O
F
O
5th choice
S
R
R
The Plurality Method
37
The Condorcet Criterion
If candidate X is preferred by the voters over each
of the other candidates in a head-to-head
comparison, then candidate X should be the winner
of the election.
Not every election has a Condorcet candidate but if there is
one, it is a good sign that this candidate represents the voice
of the voters better than any other candidate.
The Plurality Method
38

Consider once again the marching band bowl game
example.
# of voters
49
48
3
1st choice
R
H
F
2nd choice
H
S
H
3rd choice
F
O
S
4th choice
O
F
O
5th choice
S
R
R
RvH
RvF
RvO
RvS
HvF
HvO
HvS
FvO
FvS
OvS
The Plurality Method
39

The Hula Bowl is a Condorcet candidate but under the
plurality method, the Hula bowl is not the winner.
Therefore, the plurality method violates the Condorcet
criterion.
# of voters
49
48
3
1st choice
R
H
F
2nd choice
H
S
H
3rd choice
F
O
S
4th choice
O
F
O
5th choice
S
R
R
The Plurality Method
40
Insincere Voting (or Strategic Voting)
If we know that the candidate we really want
doesn’t have a chance of winning, then rather than
“wasting our vote” on our favorite candidate we
can cast it for a lesser choice that has a better
chance of winning the election.
The Plurality Method
41

Of the voting methods, the plurality method is most
susceptible to insincere voting. In the US,
presidential races are often decided by insincere
voters.
The Plurality Method
42
Jill Stein

This candidate was my
first choice, but I knew
that she had zero
chance of winning the
national election.
Barack Obama

On the other hand, this
was my second choice
and had a much better
chance of winning, so I
voted for him instead.
The Plurality Method
43

Back to the marching band example. If the three
people who voted for the Fiesta Bowl as their first
choice realized that it had little chance of winning
but that the Hula bowl, their second choice, had a
good chance of winning, they could change their
votes and therefore, change the outcome.
The Plurality Method
44
# of voters
49
48
3
1st choice
R
H
F
2nd choice
H
S
H
3rd choice
F
O
S
4th choice
O
F
O
5th choice
S
R
R
# of voters
49
48
3
1st choice
R
H
H
2nd choice
H
S
F
3rd choice
F
O
S
4th choice
O
F
O
5th choice
S
R
R
The Plurality Method
45

The plurality method of voting reinforces an
entrenched two-party system that often leaves
voters with no real choice. This is called Duverger’s
Law.
The Plurality Method
46



Who is the plurality method winner?
Is there a majority candidate?
Is there a Condorcet candidate?
6
3
5
8
1st Choice
D
D
A
C
2nd Choice
B
A
C
A
3rd Choice
A
B
B
D
4th Choice
C
C
D
B
The Plurality Method
47

Consider an election with 456 voters and seven
candidates. What is the smallest number of votes
that a plurality candidate could have?
48
THE MATHEMATICS OF
VOTING
1.3 The Borda Count Method
The Borda Count Method
49


In the Borda Count Method each place on a ballot is
assigned points. In an election with N candidates
we give 1 point for last place, 2 points for second
from last place, and so on.
The points for each candidate are tallied and the
candidate with the highest total is the winner. This
candidate is called the Borda winner.
The Borda Count Method
50

This method can sometimes result in a tie. In this unit,
assume that a tie can stand and will not be decided
using some other method.
The Borda Count Method
51

The Borda count method is good for taking into
account voters’ preferences beyond first choice.
Because the winner is the candidate with the best
average ranking, the winner is often a good
compromise candidate.
The Borda Count Method
52

Consider the election for the new principal at
Washington Elementary School. The School Board is
going to hire one of four finalists: Mrs. Amaro, Mr.
Burr, Mr. Castro, and Mrs. Dunbar. The winner will be
determined using the Borda count method.
# of voters
6
2
3
1st choice
A
B
C
2nd choice
B
C
D
3rd choice
C
D
B
4th choice
D
A
A
The Borda Count Method
53

Who is the winner? How do you think Mrs. Amaro
feels about the outcome?
# of voters
6
2
3
1st choice
A
B
C
2nd choice
B
C
D
3rd choice
C
D
B
4th choice
D
A
A
The Borda Count Method
54
Violations of Fairness Criterion
 From the previous example, we see that the Borda
Count method violates both the majority criterion
and the Condorcet criterion.
 Despite its flaws, experts in voting theory consider it
one of the best, if not the best, method for deciding
elections with many candidates.
The Borda Count Method
55
In defense of the Borda count method:
 Although violations of the majority criterion can occur,
they are very rare especially when there are many
candidates.
 The Condorcet criterion violation follows from the
majority criterion violation since a majority candidate
is also a Condorcet candidate. There can still be a
violation of the majority criterion but it is unlikely.
The Borda Count Method
56

The Borda count method is used for deciding many
elections, including:
 Heisman
trophy winner
 NBA Rookie of the Year
 NFL MVP
 College football polls
The Borda Count Method
57

Using the Borda Count method who won the
election?
6
3
5
8
1st Choice
D
D
A
C
2nd Choice
B
A
C
A
3rd Choice
A
B
B
D
4th Choice
C
C
D
B
The Borda Count Method
58

An election is held among three candidates (A, B, C)
using the Borda count method. There are 20 voters
If candidate A received 37 points and candidate B
received 39 points how many points did candidate
C receive?
59
THE MATHEMATICS OF
VOTING
1.4 The Plurality-with-elimination Method
Plurality-with-Elimination Method
60



There is rarely a majority with 3 or more candidates.
IRV = Instant Runoff Voting.
When a majority does not exist, removing the
candidate with the lowest first-place votes one at a
time until a majority exists is the Plurality-withElimination method.
Plurality-with-Elimination Method
61

Using a preference ballot, a voter can choose his or
her first choice and also rank the remaining choices.
We can therefore tell who a voter would vote for in a
runoff election, even if their first choice is eliminated.
# of
voters
14
10
8
4
1
1st
A
C
D
B
C
2nd
B
B
C
D
D
3rd
C
D
B
C
B
4th
D
A
A
A
A
Plurality-with-Elimination Method
62

Round 1. Count the first-place votes for each
candidate, just as you would in the plurality method.
If a candidate has a majority of first-place votes,
that candidate is the winner. Otherwise, eliminate
the candidate (or candidates if there is a tie) with
the fewest first-place votes.
Plurality-with-Elimination Method
63
Plurality-with-Elimination Method
64

Round 2. Cross out the name(s) of the candidates
eliminated from the preference and recount the
first-place votes. (Remember that when a candidate
is eliminated from the preference schedule, in each
column the candidates below it move up a spot.)
Plurality-with-Elimination Method
65

Round 2 (continued). If a candidate has a majority
of first-place votes, declare that candidate the
winner. Otherwise, eliminate the candidate with the
fewest first-place votes.
Plurality-with-Elimination Method
66
14 votes
11 votes
12 votes
Plurality-with-Elimination Method
67

Round 3, 4, etc. Repeat the process, each time
eliminating one or more candidates until there is a
candidate with a majority of first-place votes. That
candidate is the winner of the election.
Plurality-with-Elimination Method
68
14 votes
23 votes
Plurality-with-Elimination Method
69
# of
voters
93
44
10
30
42
81
1st
A
B
C
C
D
E
2nd
B
D
A
E
C
D
3rd
C
E
E
B
E
C
4th
D
C
B
A
A
B
5th
E
A
D
D
B
A
Plurality-with-Elimination Method
70

We can conclude that the plurality-with-elimination
method satisfies the Majority Criterion.
#
6
2
3
1st
A
B
C
2nd
B
C
D
3rd
C
D
B
4th
D
A
A
Plurality-with-Elimination Method
71
So what is wrong with the plurality-with-elimination
method?
The Monotonicity Criterion
If candidate X is a winner of an election and, in a
reelection, the only changes in the ballots are
changes that favor X (and only X), then X should
remain a winner of the election.
Plurality-with-Elimination Method
72


Athens, Barcelona, and Calgary are competing to
host the summer Olympics. The final decision is
made by a secret vote of the 29 members of the
Executive Council of the IOC and the winner will be
chosen using the plurality-with-elimination method.
A few days before the election, the results of a
straw poll are leaked. Based on the results, Calgary
is going to win.
Plurality-with-Elimination Method
73
Straw Poll
#
7
8
10
4
1st
A
B
C
A
2nd
B
C
A
C
3rd
C
A
B
B
Official Results – Everyone loves a winner
#
7
8
14
1st
A
B
C
2nd
B
C
A
3rd
C
A
B
Plurality-with-Elimination Method
74


The Olympic example illustrates a violation of the
monotonicity criterion.
The plurality-with-elimination method of voting also
violates the Condorcet criterion.
Plurality-with-Elimination Method
75

Using the Plurality-with-Elimination method who won
the election?
6
3
5
8
1st Choice
D
D
A
C
2nd Choice
B
A
C
A
3rd Choice
A
B
B
D
4th Choice
C
C
D
B
76
THE MATHEMATICS OF
VOTING
1.5 The Method of Pairwise Comparisons
Method of Pairwise Comparisons
77

All three methods we’ve looked at so far have
violated the Condorcet criterion. The next method
we will consider does not violate this fairness
criterion.
 Plurality
Method
 Borda Count Method
 Plurality-With-Elimination
Method of Pairwise Comparisons
78


In this method, every candidate is matched head-tohead against every other candidate. Each of these
match-ups is called a pairwise comparison.
In a pairwise comparison between X and Y, every
vote is assigned to either X or Y, the vote going to
the whichever of the two candidates is listed higher
on the ballot.
Method of Pairwise Comparisons
79



The winner gets 1 point, the loser gets 0 points and
in a tie, both get .5 points.
The winner of the election is the candidate with the
most points after all the pairwise comparisons have
been tabulated.
Ties can happen with this method and are quite
common.
Method of Pairwise Comparisons
80
#
2
6
4
1
1
4
4
1st
A B
B
C
C
D
E
2nd
D A A B
D
A
C
3rd
C C D A
A
E
D
4th
B
D E
D
B
C
B
5th
E
E
C E
E
B
A
A vs. B
A vs. C
A vs. D
A vs. E
B vs. C
B vs. D
B vs. E
C vs. D
C vs. E
D vs. E
Note: 10 Comparisons
A vs. B = 7 votes to 15 (B wins). B gets 1 point.
A vs. C = 16 votes to 6 (A wins). A gets 1 point.
Method of Pairwise Comparisons
81
So what is wrong with the method of pairwise
comparisons?
The Independence-of-Irrelevant-Alternatives
Criterion (IIA)
If candidate X is a winner of an election and in a
recount one of the non-winning candidates is
removed (or added) from the ballots, then X should
still be a winner of the election.
Method of Pairwise Comparisons
82
#
2
6
4
1st
A
B
B
2nd
D
A
A
B
D
A
D
A
A
E
E
D
B
E
E
3rd
4th
B
D
5th
E
E
1
1
4
4
D
E
D
B
B
A
Eliminate C (an irrelevant alternative) from this
election and B wins (rather than A).
Method of Pairwise Comparisons
83


Gauss’s method: Sum of Consecutive Integers
What is the sum of 1+2+3…99?
Method of Pairwise Comparisons
84
How Many Pairwise Comparisons?
In an election between 5 candidates, there were 10
pairwise comparisons.
How many comparisons will be needed for an
election having 6 candidates?
Ans. 5 + 4 + 3 + 2 + 1 = 15
Method of Pairwise Comparisons
85
The Number of Pairwise Comparisons
In an election with N candidates the total number of
pairwise comparisons between candidates is an
abbreviated Sum of Consecutive Integers formula:
(N -1)N
2
Method of Pairwise Comparisons
86

Using the Pairwise Comparisons method who won
the election?
6
3
5
8
1st Choice
D
D
A
C
2nd Choice
B
A
C
A
3rd Choice
A
B
B
D
4th Choice
C
C
D
B
Method of Pairwise Comparisons
87


An election is held among six candidates. What is
the total number of Pairwise Comparisons in this
election?
In a round robin tennis tournament, every player
plays against every other player. If 24 players are
entered in the tournament, how many matches will
be played?
Method of Pairwise Comparisons
88

1 + 2 + 3 + … + 99 + 100 =

2 + 4 + 6 + … + 198 + 200 =
89
THE MATHEMATICS OF
VOTING
1.6 Rankings
Rankings
90


Often, it is important to not only know who wins an
election but also to know who comes in 2nd, 3rd,
4th, etc.
Consider the Math Club election. Suppose that
instead of having multiple elections, they are
holding a single election for a board of directors,
including a president, a vice president, and a
treasurer. 1st place will be president, 2nd place will
be vice president, and 3rd place will be treasurer.
Rankings
91
Extended Ranking




Extended Plurality
Extended Borda Count
Extended Plurality with
Elimination
Extended Pairwise
Comparisons
Recursive Ranking


Recursive Plurality
Recursive Plurality with
Elimination
Rankings: Extended Plurality
92

In the extended plurality method, we count the first
place votes of each candidate and rank them in
order from most first place votes to fewest first
place votes.
#
14
10
8
4
1
1st
A
C
D
B
C
2nd
B
B
C
D
D
3rd
C
D
B
C
B
4th
D
A
A
A
A
nd: the
rd: D, 4th: B
Find the ranking
1st: A, 2of
C, 3candidates
Rankings: Extended Borda Count
93

In the extended Borda count method, the point totals
are tallied for each candidate and then the
candidates are ranked in order from highest total to
lowest total.
#
14
10
8
4
1
1st
A
C
D
B
C
2nd
B
B
C
D
D
3rd
C
D
B
C
B
4th
D
A
A
A
A
rd: D(81), 4th: A(79)
the ranking
of the3candidates
1st:Find
B(106),
2nd: C(104),
Rankings: Extended Plurality-withElimination
94

In the extended plurality-with-elimination method,
candidates are ranked by the order in which they are
eliminated. The winner is determined the usual way.
Second place is the last eliminated candidate.
#
14
10
8
4
1
1st
A
C
D
B
C
2nd
B
B
C
D
D
3rd
C
D
B
C
B
4th
D
A
A
A
A
st: D, 2nd
rd: C, 4th: B
Find the 1ranking
of: A,
the3candidates
Rankings: Extended Method of Pairwise
Comparisons
95

In the extended method of pairwise comparisons
method, candidates are ranked by point totals after
tabulating all the pairwise comparisons, highest to
lowest.
#
14
10
8
4
1
1st
A
C
D
B
C
2nd
B
B
C
D
D
3rd
C
D
B
C
B
4th
D
A
A
A
A
st: C, 2nd
Find the 1ranking
of: B,
the3rd
candidates
: D, 4th: A
Rankings: Recursive
96


A somewhat more involved strategy for ranking is
recursive ranking.
In the case of an election, the process is to find the
winner then remove the winner’s name from the
preference schedule, thus creating a new preference
schedule.
Rankings: Recursive
97
Recursive Ranking
 Step 1: [Determine first place]
Choose winner using method and remove that
candidate.
 Step 2: [Determine second place]
Choose winner of new election (without candidate
removed in step 1) and remove that candidate.
 Steps 3, 4, etc.: [Determine third, fourth, etc. places]
Continue in same manner using method on remaining
candidates yet to be ranked.
Rankings: Recursive Plurality
98
First-place: A
Second-place: B
Third-place: C
Fourth-place: D
Rankings: Recursive
99

While interesting, recursive ranking methods have
little practical use and in reality, extended ranking
methods are almost always used.
The Mathematics of Voting
Conclusion
100
Methods of Vote Counting
 Fairness Criteria
 Arrow’s Impossibility Theorem
It is mathematically impossible for a
democratic voting method to satisfy all of the
fairness criteria.
