Copyright © 2009 Pearson Education, Inc. Chapter 15 Section 2 - Slide 1

5-2 Election Theory

Flaws of Voting

Chapter 15 Section 2 - Slide 2Copyright © 2009 Pearson Education, Inc.

WHAT YOU WILL LEARN

• Flaws of voting methods

Chapter 15 Section 2 - Slide 3Copyright © 2009 Pearson Education, Inc.

Fairness Criteria

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

Chapter 15 Section 2 - Slide 4Copyright © 2009 Pearson Education, Inc.

Majority Criterion

If a candidate receives a majority (more than 50%) of the first-place votes, that candidate should be declared the winner.

Chapter 15 Section 2 - Slide 5Copyright © 2009 Pearson Education, Inc.

Head-to-Head Criterion

If a candidate is favored when compared head-to-head with every other candidate, that candidate should be declared the winner.

Chapter 15 Section 2 - Slide 6Copyright © 2009 Pearson Education, Inc.

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.

Chapter 15 Section 2 - Slide 7Copyright © 2009 Pearson Education, Inc.

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.

Chapter 15 Section 2 - Slide 8Copyright © 2009 Pearson Education, Inc.

Summary of the Voting Methods and Whether They Satisfy the Fairness Criteria

May not satisfy

May not satisfy

May not satisfy

May not satisfy

Irrelevant alternatives

Always satisfies

May not satisfy

Always satisfies

Always satisfies

Monotonicity

Always satisfies

May not satisfy

May not satisfy

May not satisfy

Head-to-head

Always satisfies

Always satisfies

May not satisfy

Always satisfies

Majority

Pairwise comparison

Plurality with elimination

Borda count

PluralityMethod

Criteria

Chapter 15 Section 2 - Slide 9Copyright © 2009 Pearson Education, Inc.

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

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Which voting method(s) – plurality, Borda count, plurality with elimination, or pairwise comparison – violate the majority criterion using the following election data?

a. Plurality b. Plurality with elimination

c. Borda count d. Pairwise comparison

Number of Votes 10 15 20 20

First A B C B

Second C A A C

Third B C B A

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Which voting method(s) – plurality, Borda count, plurality with elimination, or pairwise comparison – violate the majority criterion using the following election data?

a. Plurality b. Plurality with elimination

c. Borda count d. Pairwise comparison

Number of Votes 10 15 20 20

First A B C B

Second C A A C

Third B C B A

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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?

a. Yes b. No c. Can’t determine

Number of Votes 23 20 17 15

First B E W E

Second E W E B

Third W B B W

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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?

a. Yes b. No c. Can’t determine

Number of Votes 23 20 17 15

First B E W E

Second E W E B

Third W B B W

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Practice Problems

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