1. © 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 14 Voting and Apportionment

2. © 2010 Pearson Prentice Hall. All rights reserved. 2 14.1 Voting Methods

3. © 2010 Pearson Prentice Hall. All rights reserved. 3 Objective 1.Understand and use preference tables. 2.Use the plurality method to determine an election’s winner. 3.Use the Borda count method to determine an election’s winner. 4.Use the plurality-with-elimination method to determine an election’s winner. 5.Use the pairwise comparison method to determine an election’s winner.

4. © 2010 Pearson Prentice Hall. All rights reserved. 4 Preference Tables Preference ballots are ballots in which a voter is asked to rank all the candidates in order of preference. Example: Four candidates are running for president of the Student Film Institute: Paul (P), Rita (R), Sarah (S), and Tim (T). Each of the club’s members submits a secret ballot indicating his or her first, second, third, and fourth choice for president. The 37 ballots are shown to the right.

5. © 2010 Pearson Prentice Hall. All rights reserved. 5 A preference table shows how often each particular outcome occurred. Example: Preference Tables The election ballots are placed into identical stacks. Then the identical stacks are placed into a preference table.

6. © 2010 Pearson Prentice Hall. All rights reserved. 6 How many people selected Donna (D) as their first choice? Solution: We find the number of people who voted for D as their first choice by reading across the row that says First Choice. When you see a D in this row, write the number above it. Then find the sum of the numbers: 120 + 100 = 220 Thus, 220 people selected Donna as their first choice. Example 1: Understanding a Preference Table

7. © 2010 Pearson Prentice Hall. All rights reserved. 7 The Plurality Method The candidate (or candidates, if there is more than one) with the most first-place votes is the winner. Example: For the previous preference table, who is declared the winner using the plurality method? Solution: The candidate with the most first-place votes is the winner. When using the preference table, we only need to look at the row indicating the number of first-choice votes. Thus, P (Paul) is the winner and the new president of the Student Film Institute.

8. © 2010 Pearson Prentice Hall. All rights reserved. 8 The Borda Count Method Each voter ranks the candidates from the most favorable to the least favorable. 1.Each last-place vote is given 1 point, each next- to-last-place vote is given 2 points, each third- from-last-place vote is given 3 points, and so on. 2.The points are totaled for each candidate separately. 3.The candidate with the most points is the winner.

9. © 2010 Pearson Prentice Hall. All rights reserved. 9 Example: In the previous table, who is the winner using the Borda method? Solution: Because there are 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. Example 3: Using the Borda Count Method Number of Votes 1410841 First Choice: 4 pts P: 14  4=56S: 10  4=40T: 8  4=32R: 4  4=16S: 1  4=4 Second Choice: 3 pts R: 14  3=42R: 10  3=30S: 8  3=24T: 4  3=12T: 1  3=3 Third Choice: 2 pts S: 14  2=28T: 10  2=20R: 8  2=16S: 4  2=8R: 1  2=2 Fourth Choice: 1 pt T: 14  1=14P: 10  1=10P: 8  1=8P: 4  1=4P: 1  1=1

10. © 2010 Pearson Prentice Hall. All rights reserved. 10 Now, we read down each column and total the points for each candidate separately: P: 56 + 10 + 8 + 4 + 1 = 79 points R: 42 + 30 + 16 + 16 + 2 = 106 points S: 28 + 40 + 24 + 8 + 4 = 104 points T: 14 + 20 + 32 + 12+ 3 = 81 points Because Rita (R) has received the most points, she is the winner and the new president of the Student Film Institute. Example 3: Using the Borda Count Method

11. © 2010 Pearson Prentice Hall. All rights reserved. 11 The Plurality-with-Elimination Method The candidate with the majority of first-place votes is the winner. If no candidate receives a majority of first-place votes, eliminate the candidate (candidates, if there is a tie) with the fewest first-place votes from the preference table. Move the candidates in each column below each eliminated candidate up one place. The candidate with the majority of first-place votes in the new preference table is the winner. If no candidate receives a majority of first-place votes, repeat this process until a candidate receives a majority.

12. © 2010 Pearson Prentice Hall. All rights reserved. 12 Example: In the previous table, who is declared the winner using the plurality-with-elimination method? Example 4: Using the Plurality-with-Elimination Method Number of Votes 1410841 First ChoicePSTRS Second ChoiceRRSTT Third ChoiceSTRSR Fourth ChoiceTPPPP Solution: Recall that there are 37 people voting. In order to receive a majority, a candidate must receive more than 50% of the first-place votes, i.e., meaning 19 or more votes.

13. © 2010 Pearson Prentice Hall. All rights reserved. 13 The number of first-place votes for each candidate is P(Paul) = 14 S(Sarah) = 10 + 1 = 11 T(Tim) = 8 R(Rita) = 4 We see that no candidate receives a majority of first-place votes. Because Rita received the fewest first-place votes, she is eliminated in the first round. The new preference table is So, the number of first-place votes for each candidate is P(Paul) = 14 S(Sarah) = 10 + 1 = 11 T(Tim) = 8 + 4 = 12. Number of Votes 1410841 First ChoicePSTTS Second ChoiceSTSST Third ChoiceTPPPP Example 4: Using the Plurality-with-Elimination Method

14. © 2010 Pearson Prentice Hall. All rights reserved. 14 Once again, no candidate receives a majority first-place votes. Because Sarah received the fewest first-place votes, she is eliminated from the second round. The new preference table is The number of first-place votes for each candidate is now P(Paul) = 14 T(Tim) = 10+ 8 + 4 + 1 = 23. Because T(Tim) has received the majority of first-place votes, i.e., Tim received more than 19 votes, he is the winner and the new president of the Student Film Institute. Number of Votes 1410841 First ChoicePTTTT Second ChoiceTPPPP Example 4: Using the Plurality-with-Elimination Method

15. © 2010 Pearson Prentice Hall. All rights reserved. 15 The Pairwise Comparison Method Using the pairwise comparison method, every candidate is compared one-on-one with every other candidate. The number of comparisons made using the pairwise comparison method is given by where n is the number of candidates, and C is the number of comparisons that must be made.

16. © 2010 Pearson Prentice Hall. All rights reserved. 16 Voters rank all the candidates and the results are summarized in a preference table. The table is used to make a series of comparisons in which each candidate is compared to each of the other candidates. For each pair of candidates, X and Y, use the table to determine how many voters prefer X to Y and vice versa. If a majority prefer X to Y, then X receives 1 point. If a majority prefers Y to X, then Y receives 1 point. If the candidates tie, then each receives half a point. After all comparisons have been made, the candidate receiving the most points is the winner. The Pairwise Comparison Method

17. © 2010 Pearson Prentice Hall. All rights reserved. 17 For the previous table, who is declared the winner using the pairwise comparison method? Solution: We first find how many comparisons that must be made. Since there are 4 candidates, then n=4 and Example 5: Using the Pairwise Comparison Method Number of Votes 1410841 First ChoicePSTRS Second ChoiceRRSTT Third ChoiceSTRSR Fourth ChoiceTPPPP

18. © 2010 Pearson Prentice Hall. All rights reserved. 18 With P, R, S, T, the comparisons are P vs. R, P vs. S, P vs. T, R vs. S, R vs. T, and S vs. T. Example 5: Using the Pairwise Comparison Method

19. © 2010 Pearson Prentice Hall. All rights reserved. 19 Using the six comparisons and conclusion to add points we get P: no points S: 1 + 1 + 1 = 3 points R: 1 + 1 = 2 points T: 1 point After all the comparisons have been made, the candidate receiving the most points is S(Sarah). Sarah is the winner and new president of the Student Film Institute. Example 5: Using the Pairwise Comparison Method

20. © 2010 Pearson Prentice Hall. All rights reserved. Homework Pg 777 – 778, #1 – 6, 8, 9, 12, 13, 16, 17, 21, 22, 24, 25, 27 – 30.