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

© 2010 Pearson Prentice Hall. All rights reserved. 1

§14.2, Flaws of Voting Methods


Learning Targets

I will use the majority criterion to determine a voting system’s fairness.

I will use the head-to-head criterion to determine a voting system’s fairness.

I will use the monotonicity criterion to determine a voting system’s fairness.

I will use the irrelevant alternatives criterion to determine a voting system’s fairness.

I will understand Arrow’s Impossibility Theorem.


The Majority Criterion

If a candidate receives a majority of first-place votes in an election, then that candidate should win the election. This is called the Majority Criterion.


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

6 3 2

First Choice E G F

Second Choice F H G

Third Choice G F H

Fourth Choice H E E


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.


Number of Votes

6 3 2

First Choice E G F

Second Choice F H G

Third Choice G F H

Fourth Choice H E E


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 6 3 2

First Choice: 4 pts E: 64=24 G: 34=12 F: 24=8

Second Choice: 3 pts F: 63=18 H: 33=9 G: 23=6

Third Choice: 2 pts G: 62=12 F: 32=6 H: 22=4

Fourth Choice: 1 pt H: 61=6 E: 31=3 E: 21=2



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.



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.


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 8 6 4 4

First Choice A C C B

Second Choice B B A A

Third Choice C A B C


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.



First Choice A C C B

Second Choice B B A A

Third Choice C A B C


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.



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. This is the Monotonicity Criterion.


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)20 16 14 8

First Choice W S G G

Second Choice S W S W

Third Choice G G W S

Number of Votes(Second Election)

28 16 14

First Choice W S G

Second Choice G W S

Third Choice S G W


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?



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)

20 16 14 8

First Choice W W G G

Second Choice G G W W



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)

28 16 14

First Choice W S S

Second Choice S W W



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. Since the changes did not benefit her, this violates the monotonicity criterion.



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. This is the Irrelevant Alternatives Criterion.


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 resultsare 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?


First Choice E G H H

Second Choice F F E E

Third Choice G H G F

Fourth Choice H E F G


a. Because there are four candidates, n = 4, then the number of comparisons we must make is…

Next, we make 6 comparisons.

.6

2

144

2

1

nnC



Comparison Vote Results Conclusion

E vs. F260 voters prefer E to F.100 voters prefer F to E.

E wins and gets 1 point.

E vs. G260 voters prefer E to G.100 voters prefer G to E.

E wins and gets 1 point.

E vs. H160 voters prefer E to H.200 voters prefer H to E.

H wins and gets 1 point.

F vs. G180 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. H260 voters prefer F to H.100 voters prefer H to F.

F wins and gets 1 point.

G vs. H260 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.



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.

.1

2

122

2

1

nnC



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.



The Search for a Fair Voting SystemArrow’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.

•Homework: pg 788 – 789, #5 – 14.

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

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