1. Copyright © 2009 Pearson Education, Inc. Chapter 15 Section 2 - Slide 1 5-2 Election Theory Flaws of Voting

2. Chapter 15 Section 2 - Slide 2 Copyright © 2009 Pearson Education, Inc. WHAT YOU WILL LEARN Flaws of voting methods

3. Chapter 15 Section 2 - Slide 3 Copyright © 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

4. Chapter 15 Section 2 - Slide 4 Copyright © 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.

5. Chapter 15 Section 2 - Slide 5 Copyright © 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.

6. Chapter 15 Section 2 - Slide 6 Copyright © 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.

7. Chapter 15 Section 2 - Slide 7 Copyright © 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.

8. Chapter 15 Section 2 - Slide 8 Copyright © 2009 Pearson Education, Inc. Summary of the Voting Methods and Whether They Satisfy the Fairness Criteria May not satisfy Irrelevant alternatives Always satisfies May not satisfy Always satisfies Monotonicity Always satisfies May not satisfy Head-to- head Always satisfies May not satisfy Always satisfies Majority Pairwise comparison Plurality with elimination Borda count PluralityMethod Criteria

9. Chapter 15 Section 2 - Slide 9 Copyright © 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

10. 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? a. Pluralityb. Plurality with elimination c. Borda count d. Pairwise comparison Number of Votes101520 FirstABCB SecondCAAC ThirdBCBA

11. Slide 15 - 11 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? a. Pluralityb. Plurality with elimination c. Borda count d. Pairwise comparison Number of Votes101520 FirstABCB SecondCAAC ThirdBCBA

12. Slide 15 - 12 Copyright © 2009 Pearson Education, Inc. 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. Yesb. Noc. Can’t determine Number of Votes23201715 FirstBEWE SecondEWEB ThirdWBBW

13. Slide 15 - 13 Copyright © 2009 Pearson Education, Inc. 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. Yesb. Noc. Can’t determine Number of Votes23201715 FirstBEWE SecondEWEB ThirdWBBW

14. Slide 15 - 14 Copyright © 2009 Pearson Education, Inc. Practice Problems

15. Slide 15 - 15 Copyright © 2009 Pearson Education, Inc.

16. Slide 15 - 16 Copyright © 2009 Pearson Education, Inc.