14 1 Voting Methods 2010 Pearson Prentice Hall

14. 1 Voting Methods © 2010 Pearson Prentice Hall. All rights reserved. 1

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

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. 3

Preference Tables • A preference table shows how often each particular outcome occurred. Example: 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. 4

Example 1: Understanding a Preference Table 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. © 2010 Pearson Prentice Hall. All rights reserved. 5

Define Terms Look up the following terms and define them on a sheet of paper. Plurality Voting Method: Borda Count Voting Method: Plurality-with-Elimination Method: Pairwise Comparison Method: © 2010 Pearson Prentice Hall. All rights reserved.

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. 7

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-fromlast-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. 8

Example 3: Using the Borda Count Method 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. Number of Votes 14 10 8 First Choice: 4 pts P: 14 4=56 S: 10 4=40 T: 8 4=32 R: 4 4=16 S: 1 4=4 Second Choice: 3 pts R: 14 3=42 R: 10 3=30 S: 8 3=24 T: 4 3=12 T: 1 3=3 Third Choice: 2 pts S: 14 2=28 T: 10 2=20 R: 8 2=16 S: 4 2=8 R: 1 2=2 Fourth Choice: 1 pt T: 14 1=14 P: 10 1=10 P: 8 1=8 P: 4 1=4 P: 1 1=1 © 2010 Pearson Prentice Hall. All rights reserved. 4 1 9

Example 3: Using the Borda Count Method 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 + 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. © 2010 Pearson Prentice Hall. All rights reserved. 10

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. 11

Example 4: Using the Plurality-with-Elimination Method Example: In the previous table, who is declared the winner using the plurality-with-elimination method? Number of Votes 14 10 8 4 1 First Choice P S T R S Second Choice R R S T T Third Choice S T R S R Fourth Choice T P P 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. 12

Example 4: Using the Plurality-with-Elimination Method 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 Number of Votes 14 10 8 4 1 First Choice P S T T S Second Choice S T S S T Third Choice T P P So, the number of first-place votes for each candidate is P(Paul) = 14 S(Sarah) = 10 + 1 = 11 T(Tim) = 8 + 4 = 12. © 2010 Pearson Prentice Hall. All rights reserved. 13

Example 4: Using the Plurality-with-Elimination Method 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 Number of Votes 14 10 8 4 1 First Choice P T T Second Choice T P P 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. © 2010 Pearson Prentice Hall. All rights reserved. 14

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. 15

The Pairwise Comparison Method • 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. © 2010 Pearson Prentice Hall. All rights reserved. 16

Example 5: Using the Pairwise Comparison Method For the previous table, who is declared the winner using the pairwise comparison method? Number of Votes 14 10 8 4 1 First Choice P S T R S Second Choice R R S T T Third Choice S T R S R Fourth Choice T P P Solution: We first find how many comparisons that must be made. Since there are 4 candidates, then n=4 and © 2010 Pearson Prentice Hall. All rights reserved. 17

Example 5: Using the Pairwise Comparison Method 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. © 2010 Pearson Prentice Hall. All rights reserved. 18

Example 5: Using the Pairwise Comparison Method Using the six comparisons and conclusion to add points we get P: no points R: 1 + 1 = 2 points S: 1 + 1 = 3 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. © 2010 Pearson Prentice Hall. All rights reserved. 19

Pg 777: 1 – 37 odd © 2010 Pearson Prentice Hall. All rights reserved.