1. Section 1.1, Slide 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 11.2, Slide 1 11 Voting Using Mathematics to Make Choices.

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 11.2, Slide 2 Defects in Voting Methods 11.2 Understand the majority criterion and how it is not satisfied by the Borda count method. See how the plurality method violates Condorcet’s criterion.

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 11.2, Slide 3 Defects in Voting Methods 11.2 Recognize that the pairwise comparison method can fail to satisfy the independence-or- irrelevant-alternatives criterion. See how the plurality-with- elimination method fails the monotonicity criterion.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 4 The Majority Criterion It is clear that the plurality method and the pairwise comparison method satisfy the majority criterion.

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 5 Example: A 3-person panel is voting for car of the year from the (A)udi A8, the (B)MW Z4, the (C)orvette Z06, and the (D)odge Viper. The panel uses the Borda count method. Preference ballots for the panel members are shown. Determine the winner, and show that this outcome violates the majority criterion. (continued on next slide) The Majority Criterion

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 6 Solution: The Dodge Viper wins by the Borda count method, but the majority of first-place votes were cast for the Corvette Z06. The Majority Criterion

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 7 Condorcet’s Criterion

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 8 Example: A 7-member committee is voting using preference ballots to decide which one of (H)ungary, (I)ndia, and (T)aiwan will host an event. The partially completed preference ballots are as follows. Using the plurality voting method, H is the winner. Complete the ballots so that in head-to- head voting I defeats both H and T. (continued on next slide) Condorcet’s Criterion

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 9 Solution: Because H defeats I in the first three ballots, we must force I to defeat H on the remaining four ballots. At this point, I defeats H by 4 to 3. Condorcet’s Criterion (continued on next slide)

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 10 We want I to also defeat T in head-to-head voting. Because T is currently leading I by 2 to 0, if we make T the last choice on the uncompleted ballots, we force I to defeat T. H wins the election using the plurality method, but I defeats both H and T in head-to-head competition, so Condorcet’s criterion is not satisfied. Condorcet’s Criterion

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 11 The Independence-of-Irrelevant- Alternatives Criterion

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 12 Example: A board is voting on ways to raise taxes: increase taxes on (H)otel rooms, on (A)lcohol, or on (G)asoline. The board will use the plurality method to make its decision. The results of the vote follow. Is the independence- of-irrelevant-alternatives criterion satisfied? (continued on next slide) The Independence-of-Irrelevant- Alternatives Criterion

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 13 Solution: Suppose the board removes the tax on hotel rooms as an option, we get the following. We see that the gasoline tax now wins by a vote of 12 to 8; thus, the independence-of-irrelevant- alternatives criterion is not satisfied. The Independence-of-Irrelevant- Alternatives Criterion

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 14 Example: The following table summarizes the preference ballots cast for candidates A, B, C, and D: Determine the winner of this election (continued on next slide) The Independence-of-Irrelevant- Alternatives Criterion using pairwise comparison. Do the results of the election change if any of the losing candidates are removed?

15. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 15 Solution: We note that A is the winner in a head-to-head vote. The Independence-of-Irrelevant- Alternatives Criterion (continued on next slide)

16. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 16 Removing B and C, we see that D defeats A by 10 votes to 8. This method does not satisfy the independence-of-irrelevant-alternatives criterion. The Independence-of-Irrelevant- Alternatives Criterion

17. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 17 The Monotonicity Criterion

18. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 18 Example: An election for president of a club has (C)hang, (K)wami, and (W)oytek as candidates. Plurality-with-elimination is being used to determine the winner. Three supporters of W, who had preferred C, decide to support her in the election. W tells the new supporters to vote for C instead. If the three voters indicated in the highlighted column in the table (next slide) change their votes to W first, C second, and K third, why should this cause W concern? (continued on next slide) The Monotonicity Criterion

19. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 19 Solution: K has the fewest 1 st place votes and is eliminated. W wins the election. (continued on next slide) The Monotonicity Criterion

20. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 20 Now consider the situation if the three voters had changed their votes. In this case, C has the least votes and is eliminated. The Monotonicity Criterion (continued on next slide)

21. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 21 With C eliminated, K now wins the election. The Monotonicity Criterion

22. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.2, Slide 22 The Monotonicity Criterion