1. © 2010 Pearson Prentice Hall. All rights reserved. 1 14.2 Flaws of Voting Methods

2. © 2010 Pearson Prentice Hall. All rights reserved. 2 Objectives 1.Use the majority criterion to determine a voting system’s fairness. 2.Use the head-to-head criterion to determine a voting system’s fairness. 3.Use the monotonicity criterion to determine a voting system’s fairness. 4.Use the irrelevant alternatives criterion to determine a voting system’s fairness. 5.Understand Arrow’s Impossibility Theorem.

3. © 2010 Pearson Prentice Hall. All rights reserved. 3 The Majority Criterion If a candidate receives a majority of first-place votes in an election, then that candidate should win the election.

4. © 2010 Pearson Prentice Hall. All rights reserved. 4 Example 1: The Borda Count Method Violates the Majority Criterion The 11 members of the Board of Trustees of your college must hire a new college president. The four finalists for the job, E, F, G, and H, are ranked by the 11 members. The preference table is shown. The board members agree to use the Borda count method to determine the winner. a.Which candidate has a majority of first-place votes? b.Which candidate is declared the new college president using the Borda method? Number of Votes 632 First ChoiceEGF Second ChoiceFHG Third ChoiceGFH Fourth ChoiceHEE

5. © 2010 Pearson Prentice Hall. All rights reserved. 5 Solution: a.Since there are 11 voters, the majority has to be at least 6 votes. The first-choice row shows that candidate E received 6 first-place votes. Thus, E has the majority first-place votes and should be the new college president. Example 1: The Borda Count Method Violates the Majority Criterion Number of Votes 632 First ChoiceEGF Second ChoiceFHG Third ChoiceGFH Fourth ChoiceHEE

6. © 2010 Pearson Prentice Hall. All rights reserved. 6 b.Using the Borda method with four candidates, a first- place vote is worth 4 points, a second place vote is worth 3 points, a third-place vote is worth 2 points, and a fourth-place vote is worth 1 point. Number of Votes 632 First Choice: 4 pts E: 6  4=24G: 3  4=12F: 2  4=8 Second Choice: 3 pts F: 6  3=18H: 3  3=9G: 2  3=6 Third Choice: 2 pts G: 6  2=12F: 3  2=6H: 2  2=4 Fourth Choice: 1 pt H: 6  1=6E: 3  1=3E: 2  1=2 Example 1: The Borda Count Method Violates the Majority Criterion

7. © 2010 Pearson Prentice Hall. All rights reserved. 7 Read down the columns and total the points for each candidate. E: 24 + 3 + 2 = 29 points F: 18 + 6 + 8 = 32 points G:12 + 12 + 6 = 30 points H: 6 + 9 + 4 = 19 points Because candidate F has the most points, candidate F is declared the new college president using the Borda method. Example 1: The Borda Count Method Violates the Majority Criterion

8. © 2010 Pearson Prentice Hall. All rights reserved. 8 The Head-to-Head Criterion If a candidate is favored when compared separately-that is, head-to-head, with every other candidate, then that candidate should win the election.

9. © 2010 Pearson Prentice Hall. All rights reserved. 9 Example 2: The Plurality Method Violates the Head-to-Head Criterion Twenty-two people are asked to taste-test and rank three different brands, A, B, and C of tuna fish. The results are shown in the table. a.Which brand is favored over all others using a head- to-head comparison? b.Which brand wins the taste test using the plurality method? Number of Votes 8644 First ChoiceACCB Second ChoiceBBAA Third ChoiceCABC

10. © 2010 Pearson Prentice Hall. All rights reserved. 10 Solution: a.Compare brands A and B. A is favored over B in columns 1 & 3 giving A 12 votes. B is favored over A in columns 2 & 4 giving B 10 votes. Hence, A is favored when compared to B. Example 2: The Plurality Method Violates the Head-to-Head Criterion Number of Votes 8644 First ChoiceACCB Second ChoiceBBAA Third ChoiceCABC

11. © 2010 Pearson Prentice Hall. All rights reserved. 11 a.(cont.) Compare brands A and C. A is favored over C in columns 1 & 4 giving A 12 votes. C is favored over A in columns 2 & 3 giving C 10 votes. Hence, A is favored when compared to C. Notice that A is favored over the other two brands using head-to-head comparison. b.Using the plurality method, the brand with the most first-place votes is the winner. Since A received 8 first-place votes, B received 4 votes, and C received 10 votes, so Brand C wins the taste test. Example 2: The Plurality Method Violates the Head-to-Head Criterion

12. © 2010 Pearson Prentice Hall. All rights reserved. 12 Monotonicity Criterion If a candidate wins an election and, in a reelection, the only changes are changes that favor that candidate, then that candidate should win the reelection.

13. © 2010 Pearson Prentice Hall. All rights reserved. 13 Example 3: The Plurality-with-Elimination Method Violates the Monotonicity Criterion The 58 members of the Student Activity Council are meeting to elect a keynote speaker to launch student involvement week. The choices are Bill Gates (G), Howard Stern (S), or Oprah Winfrey (W). After a straw vote, 8 students change their vote so Oprah Winfrey (W) is their first choice. Number of Votes (Straw Vote) 2016148 First ChoiceWSGG Second ChoiceSWSW Third ChoiceGGWS Number of Votes (Second Election) 281614 First ChoiceWSG Second ChoiceGWS Third ChoiceSGW

14. © 2010 Pearson Prentice Hall. All rights reserved. 14 a.Using the plurality-with-elimination method, which speaker wins the first election? b.Using the plurality-with-elimination method, which speaker wins the second election? c.Does this violate the monotonicity criterion? Example 3: The Plurality-with-Elimination Method Violates the Monotonicity Criterion

15. © 2010 Pearson Prentice Hall. All rights reserved. 15 Solution: a.Since there are 58 voters, the majority of votes has to be 30 or more votes. No speaker receives the majority vote in the first election. So, we eliminate S because S had the fewest first- place votes. The new preference table is Since W received the majority of first-place votes, 36 votes, Oprah Winfrey is the winner of the straw vote. Number of Votes (Straw Vote) 2016148 First ChoiceWWGG Second ChoiceGGWW Example 3: The Plurality-with-Elimination Method Violates the Monotonicity Criterion

16. © 2010 Pearson Prentice Hall. All rights reserved. 16 b.Again, we look for the candidate that has received the majority of the votes, 30 or more. We see that no candidate has received the majority of the votes in the second election. So, we eliminate G because G receives the fewest first-place votes. The new preference table is Because S has the majority of the votes, 30 exactly, Howard Stern is the winner of the second election. Number of Votes (Second Election) 281614 First ChoiceWSS Second ChoiceSWW Example 3: The Plurality-with-Elimination Method Violates the Monotonicity Criterion

17. © 2010 Pearson Prentice Hall. All rights reserved. 17 c.Oprah Winfrey won the first election. She then gained additional support with the eight voters who changed their ballots. However, she lost the second election. This violates the monotonicity criterion. Example 3: The Plurality-with-Elimination Method Violates the Monotonicity Criterion

18. © 2010 Pearson Prentice Hall. All rights reserved. 18 The Irrelevant Alternatives Criterion If a candidate wins an election and, in a recount, the only changes are that one or more of the other candidates are removed from the ballot, then that candidate should still win the election.

19. © 2010 Pearson Prentice Hall. All rights reserved. 19 Example 4: The Pairwise Comparison Method Violates the Irrelevant Alternatives Criterion Four candidates, E, F, G, and H are running for mayor of Bolinas. The election results are shown. a.Using the pairwise comparison method, who wins this election? b.Prior to the announcement of the election results, candidates F and G both withdraw from the running. Using the pairwise comparison method, which candidate is declared mayor of Bolinas with F and G eliminated from the preference table? c.Does this violate the irrelevant alternatives criterion? Number of Votes 1601008020 First ChoiceEGHH Second ChoiceFFEE Third ChoiceGHGF Fourth ChoiceHEFG

20. © 2010 Pearson Prentice Hall. All rights reserved. 20 a.Because there are four candidates, n = 4, then the number of comparisons we must make is Next, we make 6 comparisons. Example 4: The Pairwise Comparison Method Violates the Irrelevant Alternatives Criterion

21. © 2010 Pearson Prentice Hall. All rights reserved. 21 ComparisonVote ResultsConclusion E vs. F 260 voters prefer E to F. 100 voters prefer F to E. E wins and gets 1 point. E vs. G 260 voters prefer E to G. 100 voters prefer G to E. E wins and gets 1 point. E vs. H 160 voters prefer E to H. 200 voters prefer H to E. H wins and gets 1 point. F vs. G 180 voters prefer F to G. 180 voters prefer G to F. It’s a tie. F gets ½ point and G gets ½ point. F vs. H 260 voters prefer F to H. 100 voters prefer H to F. F wins and gets 1 point. G vs. H 260 voters prefer G to H. 100 voters prefer H to G. G wins and gets 1 point. Thus, E gets 2 points, F and G each get 1½ points, and H gets 1 point. Therefore, E is the winner. Example 4: The Pairwise Comparison Method Violates the Irrelevant Alternatives Criterion

22. © 2010 Pearson Prentice Hall. All rights reserved. 22 b.Once F and G withdraw from the running, only two candidates, E and H, remain. The number of comparisons we must make is The one comparison we need to make is E vs. H. Since 200 voters prefer H to E, H is the winner and is the new mayor of Bolinas. Example 4: The Pairwise Comparison Method Violates the Irrelevant Alternatives Criterion

23. © 2010 Pearson Prentice Hall. All rights reserved. 23 c.The first election count produced E as the winner. When F and G were withdrawn from the election, the winner was H, not E. This violates the irrelevant alternatives criterion. Example 4: The Pairwise Comparison Method Violates the Irrelevant Alternatives Criterion

24. © 2010 Pearson Prentice Hall. All rights reserved. 24 The Search for a Fair Voting System Arrow’s Impossibility Theorem It is mathematically impossible for any democratic voting system to satisfy each of the four fairness criteria. In 1951, economist Kenneth Arrow proved that there does not exist, and will never exist, any democratic voting system that satisfies all of the fairness criteria.

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