MGF 1107

SUPPLEMENTAL HOMEWORK/ CHAPTER 9

 

1. What is the minimum number of votes that constitutes a majority if the number of votes cast is:

a) 98

b) 111

c) x, where x is odd.  Express your answer as a single fraction.

d) x, where x is even.  Express your answer as a single fraction.

2. Given the following preference lists, find the winner using plurality with runoff.

10
8
6
2
A
B
C
B
C
C
B
A
B
A
A
C

3. What other voting method would follow exactly the same procedure in problem 2 as plurality with runoff?

4. The 1980 U.S. Senate race in New York used the plurality method.  The results were

            D’Amato (Rep.)           45%

            Holtzman (Dem.)          44%

            Javits (Liberal party)     11%

so D’Amato won.  Exit polls showed that two-thirds of Javits supporters preferred Holtzman to D’Amato.  If the plurality with runoff method were used instead, what percentage of the votes would D’Amato and Holtzman have received in the runoff?  Round off your answer to the nearest tenth of a percent.

5. In an election with 4 alternatives and 25 voters, what is the sum of the Borda points? 

6. Find all values of x that make C the Borda count winner but not the Condorcet winner.

10
5
4
x
A
B
C
C
B
C
A
B
C
A
B
A

7. Using the same preference lists as in #6, find all values of x that make B the Borda count winner.

8. Find all values of x for which B is the Borda count winner and D is the plurality with runoff winner.

10
8
6
4
x
A
B
D
D
C
B
C
C
B
D
C
A
A
A
B
D
D
B
C
A

 9. You are on a committee that will use sequential pairwise voting to choose from among 4 alternatives: A, B, C, and D.  Your staff has researched the positions of the other voters so you are familiar with everyone’s preference lists.  As committee chairperson, you get to decide the agenda.  You are the first of the preference lists below, and hence want D to win.  Find an agenda that will accomplish your goal.  (Hint: There are 12 possible agendas but only 3 that you should bother trying.)

1
2
1
1
2
D
D
A
C
B
A
C
B
A
A
C
B
C
B
D
B
A
D
D
C

10. An election with 4 candidates (A, B, C, and D) and 150 voters is to be decided using the plurality method.  After 120 ballots have been recorded, A has 26 votes, B has 18 votes, C has 42 votes, and D has 34 votes.

a) What is the smallest number of the remaining 30 votes that A must receive to guarantee a win (without a tie) for A?

b) What is the smallest number of the remaining 30 votes that C must receive to guarantee a win (without a tie) for C?

11. Consider the following preference lists:

8
7
6
2
1
A
D
D
C
E
B
B
B
A
A
C
A
E
B
D
D
C
C
D
B
E
E
A
E
C

a) Find the winner using a Borda count.

b) Is there a Condorcet winner and, if so, who?

c) This example illustrates that the Borda count violates which two desirable properties of a voting system?

12. Consider the following preference lists:

1
1
1
A
C
B
B
A
D
D
B
C
C
D
A

a) Who wins using sequential pairwise voting and the agenda BACD?

b) Suppose the middle voter changes his mind and reverses his ranking of A and B.  If the other two voters have unchanged preferences, who wins now using the same agenda BACD?

c) Parts (a) and (b) show that sequential pairwise voting fails to satisfy what desirable property of a voting system?

13. The National Hockey League is trying to determine which defenseman should receive the coveted Norris Trophy for best defensive player.  The candidates are Al MacInnis (A), Ray Bourque (B), Chris Chelios (C), and Eric Desjardins (D). The preference lists of the 17 voters are shown below.

2

6

4

5

A

A

B

C

D

D

A

B

B

C

D

A

C

B

C

D

a) Who wins using sequential pairwise voting and the agenda ABCD?

b) How many voters prefer A to D?

c) Parts (a) and (b) show that sequential pairwise voting fails to satisfy what desirable property of a voting system?

14. An election with 4 alternatives and 26 voters has the following preference lists.

9

6

2

4

5

W

X

Y

Y

Z

Z

Y

X

Z

X

X

Z

Z

X

Y

Y

W

W

W

W

a) Who wins under the Hare system?

b) Suppose the middle two voters change their mind and reverse their ranking of X and Y.  A new election is now held again using the Hare system.  Who wins now?

c) Parts (a) and (b) show that the Hare system fails to satisfy what desirable property of a voting system?

15. If an election has a Condorcet winner, under which voting method is the Condorcet winner always guaranteed to finish first?

16. What does Arrow’s Impossibility Theorem say?

17. Which desirable property of a voting system is satisfied by none of the voting methods we studied?
18. Suppose we have 3 voters and 4 alternatives and suppose the sequence of preference lists is as follows:

1

1

1

A

C

B

B

A

D

D

B

C

C

D

A

Show that if the voting system being used is sequential pairwise voting with a fixed agenda, and if you have agenda-setting power (i.e. you get to choose the order), then you can arrange for whichever alternative you want to win the election.  (Your answer will consist of 4 agendas, one that produces A as the winner, one that produces B as the winner, etc. )

19. Suppose the following preference lists represent the true preferences of the 17 voters involved:

7

5

4

1

A

C

B

A

B

A

C

B

C

B

A

C

a) Find the winner of the election if the Hare system is used and everyone votes sincerely.

b) Find the winner if the voter on the far right votes strategically.

20. Show that the nonmonotonicity of the Hare system can also be demonstrated by the following 17-voter, 4-alternative election. (The text uses a simpler 13-voter, 3-alternative example.)

7

5

4

1

A

C

B

D

D

A

C

B

B

B

D

A

C

D

A

C

21. Suppose we have 10 candidates (A, B, C, D, E, F, G, H, I, and J).  Exhibit 10 ballots so that if each ballot were to be held by 10% of the electorate, then 90% of the voters would prefer A to B, 90% would prefer B to C, and so on down to 90% preferring I to J and then 90% preferring J to A.  (Hint: Mimic the pattern for the three-candidate case in the text that gave us Condorcet’s voting paradox.)

22. How many different ways can a voter rank 3 choices when ties are not allowed?
23. The top five vote-getters, in terms of percentages, elected to the Baseball Hall of Fame were, in alphabetical order:

Hank Aaron, 1982406 votes from 415 ballots
George Brett, 1999488 votes from 497 ballots
Ty Cobb, 1936222 votes from 226 ballots
Nolan Ryan, 1992491 votes from 497 ballots
Tom Seaver, 1992425 votes from 430 ballots

a) Rank the players from 1 to 5 in terms of voting percentage.
b) How many more votes would the fifth place finisher have needed to take over first place?
24. A player remains on the ballot for the Baseball Hall of Fame for 15 years provided he receives 5% of the votes cast each year. Some other players (and their votes) in the 1993 election were:

Mickey Lolich (43)
Thurman Munson (40)
Rusty Staub (32)
Bill Madlock (19)
Ron Cey (8)

Which of these five players meets the 5% cutoff criterion (for the 423 votes cast) for remaining on the ballot for the 1994 election?
25. Consider the following set of preference lists:

1
1
1
1
1
1
1
C
D
C
B
E
D
C
A
A
E
D
D
E
A
E
E
D
A
A
A
E
B
C
A
E
C
B
B
D
B
B
C
B
C
D

Calculate the winner using:
a) plurality voting
b) the Borda count
c) sequential pairwise voting with the agenda A, B, C, D, E
d) the Hare system
26. An interesting variant of the Hare system was proposed by the psychologist Clyde Coombs. It operates exactly as does the Hare system, but instead of deleting alternatives with the fewest first-place votes, it deletes those with the most last-place votes.
a) Use the Coombs procedure to find the winner if the ballots are as in the previous problem.
b) Show that for two voters and three alternatives, it is possible to have ballots that result in one alternative winning if the Coombs procedure is used and a tie between the other two if the Hare system is used.
27. The 45 members of a school's football team vote on three nominees, A, B, and C, by approval voting for the award of "most improved player" as indicated in the following table. An X indicates an approval vote.

Nominee78996312
AX  XX X 
B X X XX 
C  X XXX 

a) Which nominee is selected for the award?
b) Which nominee gets announced as runner-up for the award?
c) Note that two of the players abstained, that is, approved none of the nominees. Note also that one person approved of all three of the nominees. What would be the difference in the outcome if one were to "abstain" or "approve of everyone"?

28. In a few sentences, explain why:
a) plurality voting satisfies the Pareto condition.
b) plurality voting satisfies monotonicity.

c) sequenctial pairwise voting satisfies the Condorcet winner criterion.
d) sequenctial pairwise voting satisfies monotonicity.
e) the plurality with runoff method satisfies the Pareto condition.
f) the Hare system satisfies the majority criterion.
g) plurality with runoff satisfies the majority criterion.

 

Answers:

1a) 50 
1b) 56 
1c)     
1d)

2. B (C is eliminated in the first round, then B defeats A, 16-10)

3. Hare system

4. Holtzman 51.3%, D’Amato 48.7%

5. (3 + 2 + 1 + 0)(25) = 150

6. 8, 9, 10, or 11

7. 5 or 6

8. For no values of x

9. ABCD (or equivalently BACD)

10a) 24   10b) 12

11a) B    11b) Yes, D    11c) the Majority criterion and the Condorcet winner criterion

12a) D    12b) B            12c) Independence of irrelevant alternatives.

13a) D    13b) all 17       13c) the Pareto condition

14a) X    14b) Z             14c) monotonicity

15. Sequential pairwise voting

16. No voting system satisfies all the desirable properties of a voting system.

17. Independence of irrelevant alternatives
18. A wins if the agenda is BCDA, BDCA, CBAD, DCAB or CDBA. B wins if the agenda is ACDB, ACBD or ADCB. C wins if the agenda is DBAC, ADBC or ABDC. D wins if the agenda is BACD.

19a) If everyone votes sincerely, B is first eliminated and then C wins. 

19b) If the voter on the far right votes strategically, then he or she will submit a ballot with B over A over C. When the Hare system is then used, B and C are both eliminated, leaving A as the winner.

20. D is eliminated first, with B and C simultaneously eliminated in the next stage. Thus, A is the winner. Now suppose the voter on the far right moves A up. Then D is eliminated first, but only B gets eliminated in stage 2. Now A has 8 first place votes to 9 for C, so A is eliminated and C is the winner.

21.

A
J
I
H
G
F
E
D
C
B
B
A
J
I
H
G
F
E
D
C
C
B
A
J
I
H
G
F
E
D
D
C
B
A
J
I
H
G
F
E
E
D
C
B
A
J
I
H
G
F
F
E
D
C
B
A
J
I
H
G
G
F
E
D
C
B
A
J
I
H
H
G
F
E
D
C
B
A
J
I
I
H
G
F
E
D
C
B
A
J
J
I
H
G
F
E
D
C
B
A

 

 

 

 

 

 

 

 

 

 

 

22. 6
23a) 1. Seaver (98.84%), 2. Ryan (98.79%), 3. Cobb (98.23%), 4. Brett (98.19%), 5. Aaron (97.83%)
23b) 5 more votes
24. Lolich, Munson, and Staub.

25a) C25b) E25c) E25d) D


26a) E wins
26b)
A
C
B
B
C
A
27a) A27b) B27c) There would be no difference in the ranking of the nominees


28a) Suppose everyone prefers A to B. B cannot have a plurality since it cannot have any first place votes. Hence, B doesn't win.
28b) Assume alternative A has the most first place votes. Any change that improves A's ranking cannot subtract first place votes from A or add first place votes to another alternative. Hence A still has a plurality.
28c) Assume alternative A is the Condorcet winner. Then A beats every other alternative in a pairwise comparison. So A will win a sequential pairwise vote regardless of agenda.
28d) Suppose alternative A is the winner under sequential pairwise voting. Suppose the only change is that one voter puts A above B. The only head-to-head pairing affected is A vs. B. If this pairing occurs in the agenda, A still wins.
28e) Suppose everyone prefers A to B. Since B cannot have any first place votes, B cannot make it into a runoff. Hence, B doesn't win.
28f) Suppose A has a majority of the first place votes. Then A can never have the fewest first place votes. Hence, A is never eliminated and wins.
28g) Suppose A has a majority of the first place votes. Then A wins without a runoff.