Copyright © 2009 Pearson Education, Inc. Chapter 15 Section 1 - Slide 1 5-1 Election Theory Voting Methods

Copyright © 2009 Pearson Education, Inc. Chapter 15 Section 1 - Slide 1

5-1 Election Theory

Voting Methods

Chapter 15 Section 1 - Slide 2Copyright © 2009 Pearson Education, Inc.

WHAT YOU WILL LEARN• Preference tables• Voting methods


Example: Voting

Voting for Math Club President: Four students are running for president of the Math Club: Jerry, Thomas, Annette and Becky. The club members were asked to rank all candidates. The resulting preference table for this election is shown on the next slide.a) How many students voted in the election?b) How many students selected the candidates

in this order: A, J, B, T?c) How many students selected A as their first choice?


Example: Voting (continued)

a) How many students voted in the election?Add the row labeled Number of Votes14 + 12 + 9 + 4 + 1 = 40Therefore, 40 students voted in the election.

TTTTBFourthABJ

4

JBJAThirdBJAJSecondAABTFirst

191214# of Votes



b) How many students selected the candidates in this order: A, J, B, T?3rd column of numbers, 9 people

TTTTBFourthABJ

4


191214# of Votes



c) How many students selected A as their first choice?9 + 1 = 10

TTTTBFourthABJ

4


191214# of Votes


Plurality Method

This is the most commonly used method, and it is the easiest method to use when there are more than two candidates.

Each voter votes for one candidate. The candidate receiving the most votes is declared the winner.


Example: Plurality Method

Who is elected math club president using the plurality method?

We will assume that each member would vote for the person he or she listed in first place.

TTTTBFourthABJ

4


191214# of Votes


Example: Plurality Method (continued)

Thomas would be elected since he received the most votes. Note that Thomas received 14/40, or 35%, of the first-place votes, which is less than a majority.

Thomas received 14 votes Becky received 12 votes Annette received 10 votes Jerry received 4 votes


Borda Count Method

Voters rank candidates from the most favorable to the least favorable. Each last-place vote is awarded one point, each next-to-last-place vote is awarded two points, each third-from-last-place vote is awarded three points, and so forth. The candidate receiving the most points is the winner of the election.


Example: Borda Count

Use the Borda count method to determine the winner of the election for math club president.

Since 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: Borda Count (continued)

Thomas 14 first place votes 0 second place 0 third place 26 fourth place 14(4) + 0 + 0 + 26(1) = 82

Annette 10 first place votes 12 second place 18 third place 0 fourth place 10(4) + 12(3) + 18(2) + 0

= 112

TTTTBFourthABJ

4


191214# of Votes



Betty 12 first place votes 5 second place 9 third place 14 fourth place 12(4) + 5(3) + 9(2) + 14 =

95

Jerry 4 first place votes 23 second place 13 third place 0 fourth place 4(4) + 23(3) + 13(2) + 0 =

111

TTTTBFourthABJ

4


191214# of Votes



Thomas - 82 Annette - 112 Betty - 95 Jerry - 111 Annette, with 112 points, receives the most

points and is declared the winner.


Plurality with Elimination

Each voter votes for one candidate. If a candidate receives a majority of votes, that candidate is declared the winner. If no candidate receives a majority, eliminate the candidate with the fewest votes and hold another election. (If there is a tie for the fewest votes, eliminate all candidates tied for the fewest votes.) Repeat this process until a candidate receives a majority.


Example: Plurality with Elimination

Use the plurality with elimination method to determine the winner of the election for president of the math club.

Count the number of first place votes Annette 10 Betty 12 Thomas 14 Jerry 4

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4


191214# of Votes


Example: Plurality with Elimination (continued) Since 40 votes were cast, a candidate must

have 20 first place votes to receive a majority. Jerry had the fewest number of first place votes, so he is eliminated.

Redo the table. Thomas 14 Annette 10 Betty 16

T

A

B

4

TTTBThird

BBAASecond

AABTFirst

191214# of Votes


Example: Plurality with Elimination (continued) Still, no candidate received a majority. Annette

has the fewest number of first-place votes, so she is eliminated.

New preference table Betty 26 Thomas 14 Betty is the winner. T

B

4

TTTBSecondBBBTFirst

191214# of Votes


Pairwise Comparison Method

Voters rank the candidates. A series of comparisons in which each candidate is compared with each of the other candidates follows. If candidate A is preferred to candidate B, A receives one point. If candidate B is preferred to candidate A, B receives 1 point. If the candidates tie, each receives ½ point. After making all comparisons among the candidates, the candidate receiving the most points is declared the winner.


Example: Pairwise Comparison

Use the pairwise comparison method to determine the winner of the election for math club president.

Number of comparisons needed:

c

n(n 1)2

4(3)

26


Example: Pairwise Comparison (continued) Thomas versus Jerry

T = 14 J = 12 + 9 + 4 + 1 = 26 Jerry = 1 Thomas versus Annette

T = 14 A = 12 + 9 + 4 + 1 = 26 Annette = 1 Thomas versus Betty

T = 14 B = 12 + 9 + 4 + 1 = 26 Betty = 1

TTTTBFourthABJ

4


191214# of Votes


Example: Pairwise Comparison (continued) Betty versus Annette

B = 12 + 4 = 16 A = 14 + 9 + 1 = 24 Annette = 1 Betty versus Jerry

B = 12 + 1 = 13 J = 14 + 9 + 4 = 27 Jerry = 1 Annette versus Jerry

A = 12 + 9 + 1 = 22 J = 14 + 4 = 18 Annette = 1

TTTTBFourthABJ

4


191214# of VotesAnnette would win

with 3 total points, the most from the pairwise comparison method.

Slide 15 - 23Copyright © 2009 Pearson Education, Inc.

The employees at a local law firm are voting on the entrée for their annual holiday party. Their choices are chicken (C), salmon (S), and steak (T). The preference table follows.

Number of Votes 30 18 12 23

First T C T S

Second S T C C

Third C S S T


Here’s the preference table, again.How many employees voted?

a. 48 b. 58 c. 60 d. 83


First T C T S

Second S T C C

Third C S S T


Here’s the preference table, again.How many employees voted?

a. 48 b. 58 c. 60 d. 83


First T C T S

Second S T C C

Third C S S T


Here’s the preference table, again.Does any choice have a majority of votes?

a. Yes. b. No. c. Can’t determine.


First T C T S

Second S T C C

Third C S S T


Here’s the preference table, again.Does any choice have a majority of votes?

a. Yes. b. No. c. Can’t determine.


First T C T S

Second S T C C

Third C S S T


Here’s the preference table, again.Determine the winner using the plurality method.

a. Chicken b. Salmon

c. Steak d. Can’t determine


First T C T S

Second S T C C

Third C S S T


Here’s the preference table, again.Determine the winner using the plurality method.




First T C T S

Second S T C C

Third C S S T


Here’s the preference table, again. Determine the winner using the Borda count method.




First T C T S

Second S T C C

Third C S S T


Here’s the preference table, again. Determine the winner using the Borda count method.




First T C T S

Second S T C C

Third C S S T


Here’s the preference table, again. Determine the winner using the plurality method with elimination method.




First T C T S

Second S T C C

Third C S S T


Here’s the preference table, again. Determine the winner using the plurality method with elimination method.




First T C T S

Second S T C C

Third C S S T


Here’s the preference table, again. Determine the winner using the pairwise comparison method.




First T C T S

Second S T C C

Third C S S T


Here’s the preference table, again. Determine the winner using the pairwise comparison method.




First T C T S

Second S T C C

Third C S S T


The employees also must pick the dessert for their holiday party. The choices are cheese cake (C), apple tart (A), chocolate mousse cake (M), or vanilla ice cream (V). Here’s the preference table.

Number of Votes 19 16 12 22 14

First A C M V A

Second M A V M M

Third C V A C C

Fourth V M C A V


Which dessert wins this election if the plurality method is used?

a. Cheese cake b. Apple tart

c. Chocolate Mousse d. Vanilla ice cream

Number of Votes 19 16 12 22 14First A C M V A

Second M A V M M

Third C V A C C

Fourth V M C A V


Which dessert wins this election if the plurality method is used?




Second M A V M M

Third C V A C C

Fourth V M C A V


Which dessert wins this election if the Borda count method is used?




Second M A V M M

Third C V A C C

Fourth V M C A V


Which dessert wins this election if the Borda count method is used?




Second M A V M M

Third C V A C C

Fourth V M C A V


Which dessert wins this election if the plurality with elimination method is used?




Second M A V M M

Third C V A C C

Fourth V M C A V


Which dessert wins this election if the plurality with elimination method is used?




Second M A V M M

Third C V A C C

Fourth V M C A V


Which dessert wins this election if the pairwise comparison method is used?




Second M A V M M

Third C V A C C

Fourth V M C A V


Which dessert wins this election if the pairwise comparison method is used?




Second M A V M M

Third C V A C C

Fourth V M C A V


Practice Problems






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Copyright © 2009 Pearson Education, Inc. Chapter 15 Section 1 - Slide 1 5-1 Election Theory Voting Methods