Transcript No Slide Title

5-2 Election Theory
Flaws of Voting
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 2 - Slide 1
WHAT YOU WILL LEARN
• Flaws of voting methods
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 2 - Slide 2
Fairness Criteria
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Mathematicians and political scientists have
agreed that a voting method should meet the
following four criteria in order for the voting
method to be considered fair.
Majority Criterion
Head-to-head Criterion
Monotonicity Criterion
Irrelevant Alternatives Criterion
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Chapter 15 Section 2 - Slide 3
Majority Criterion

If a candidate receives a majority (more than
50%) of the first-place votes, that candidate
should be declared the winner.
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Chapter 15 Section 2 - Slide 4
Head-to-Head Criterion

If a candidate is favored when compared headto-head with every other candidate, that
candidate should be declared the winner.
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Chapter 15 Section 2 - Slide 5
Monotonicity Criterion

A candidate who wins a first election and then
gains additional support without losing any of
the original support should also win a second
election.
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Chapter 15 Section 2 - Slide 6
Irrelevant Alternatives Criterion

If a candidate is declared the winner of an
election and in a second election one or more of
the other candidates is removed, the previous
winner should still be declared the winner.
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Chapter 15 Section 2 - Slide 7
Summary of the Voting Methods and Whether
They Satisfy the Fairness Criteria
Borda
count
Plurality
Pairwise
with
comparison
elimination
Always
satisfies
May not
satisfy
Always
satisfies
Always
satisfies
May not
satisfy
May not
satisfy
May not
satisfy
Always
satisfies
Monotonicity Always
satisfies
Always
satisfies
May not
satisfy
Always
satisfies
Irrelevant
alternatives
May not
satisfy
May not
satisfy
May not
satisfy
Method Plurality
Criteria
Majority
Head-tohead
May not
satisfy
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Chapter 15 Section 2 - Slide 8
Arrow’s Impossibility Theorem

It is mathematically impossible for any
democratic voting method to simultaneously
satisfy each of the fairness criteria:
 The majority criterion
 The head-to-head criterion
 The monotonicity criterion
 The irrevelant alternative criterion
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 2 - Slide 9
Which voting method(s) – plurality, Borda count,
plurality with elimination, or pairwise comparison
– violate the majority criterion using the following
election data?
Number of Votes
10
15
20
20
First
A
B
C
B
Second
C
A
A
C
Third
B
C
B
A
a. Plurality
elimination
b. Plurality with
c. Borda count
d. Pairwise comparison
Slide 15 - 10
Copyright © 2009 Pearson Education, Inc.
Which voting method(s) – plurality, Borda count,
plurality with elimination, or pairwise comparison
– violate the majority criterion using the following
election data?
Number of Votes
10
15
20
20
First
A
B
C
B
Second
C
A
A
C
Third
B
C
B
A
a. Plurality
elimination
c. Borda count
Copyright © 2009 Pearson Education, Inc.
b. Plurality with
d. Pairwise comparison
Slide 15 - 11
The high school band is voting on a new mascot.
Their choices are a bulldog (B), an eagle (E), and
a wildcat (W). The 75 committee members rank
their choices according to the following preference
table. Does the plurality with elimination method
violate the head- to-head criterion?
Number of Votes
23
20
17
15
First
B
E
W
E
Second
E
W
E
B
Third
W
B
B
W
a. Yes
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b. No
c. Can’t determine
Slide 15 - 12
The high school band is voting on a new mascot.
Their choices are a bulldog (B), an eagle (E), and
a wildcat (W). The 75 committee members rank
their choices according to the following preference
table. Does the plurality with elimination method
violate the head- to-head criterion?
Number of Votes
23
20
17
15
First
B
E
W
E
Second
E
W
E
B
Third
W
B
B
W
a. Yes
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b. No
c. Can’t determine
Slide 15 - 13
Practice Problems
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Slide 15 - 14
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Slide 15 - 15
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Slide 15 - 16