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

© 2010 Pearson Prentice Hall. All rights reserved Voting Methods

© 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.

© 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.

© 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.

© 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: = 220 Thus, 220 people selected Donna as their first choice. Example 1: Understanding a Preference Table

© 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.

© 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.

© 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 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

© 2010 Pearson Prentice Hall. All rights reserved. 10 Now, we read down each column and total the points for each candidate separately: P: = 79 points R: = 106 points S: = 104 points T: = 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

© 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.

© 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 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.

© 2010 Pearson Prentice Hall. All rights reserved. 13 The number of first-place votes for each candidate is P(Paul) = 14 S(Sarah) = = 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) = = 11 T(Tim) = = 12. Number of Votes First ChoicePSTTS Second ChoiceSTSST Third ChoiceTPPPP Example 4: Using the Plurality-with-Elimination Method

© 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) = = 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 First ChoicePTTTT Second ChoiceTPPPP Example 4: Using the Plurality-with-Elimination Method

© 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.

© 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

© 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 First ChoicePSTRS Second ChoiceRRSTT Third ChoiceSTRSR Fourth ChoiceTPPPP

© 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

© 2010 Pearson Prentice Hall. All rights reserved. 19 Using the six comparisons and conclusion to add points we get P: no points S: = 3 points R: = 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

© 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.