Chapter 1 Review
SHORT ANSWER. Answer each question. Circle your final answer. Show all work.
Determine whether any of the listed candidates has a majority.
1) Four candidates running for congress receive votes as follows:
Alberts: 45,771, Brown: 12,483, Cassimatis: 33,288, DʹAmico: 29,127
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
2) Consider an election with 769 voters and six candidates. What is the smallest number of votes that a plurality
candidate could have?
A) 385
B) 384
C) 129
D) 128
For an election with four candidates (A, B, C, and D) we have the following preference schedule:
Number of voters
1st choice
2nd choice
3rd choice
4th choice
15
A
B
C
D
11
C
B
D
A
9 6 2
D B C
C D D
B C B
A A A
3) How many first-place votes are needed for a majority in this election?
A) 20
B) 15
C) 43
D) 22
For an election with candidates (A, B, C, D, and E), we have the following preference schedule:
Number of voters 51 48
1st choice
A D
2nd choice
B C
3rd choice
C B
4th choice
D A
5th choice
E E
5
E
C
D
B
A
4) How many people voted in the election?
A) 51
B) 104
C) 5
D) 3
SHORT ANSWER. Answer each question. Circle your final answer. Show all work.
Use the given information to answer the question.
5) Four students are running for president of the Mathemetics Club: Arthur (A), Brandy (B), Chandra (C), and
Darrell (D). The preference ballots for the candidates are shown. Construct a preference table to illustrate the
results of the voting.
DBAC DABC DBAC DBAC DABC ACBD
CABD DABC DBAC DABC ACBD DABC
1
Solve the problem.
6) If there are n = 8 candidates, how many different rankings of the candidates are there?
Determine the answer as requested.
7) The city council gave a questionnaire to its citizens, asking them to rank their priorities for next yearʹs budget.
People were asked to rank the following four choices: (P)olice, (R)oads, (S)chools, and (T)rash removal. The
results are summarized in the table below.
Number of Ballots
Preference 15 30 18 17
1st
P T P R
2nd
R R S T
3rd
S S T S
4th
T P R P
10
R
S
P
T
2
S
P
R
T
Which option is selected using the plurality method?
Use the preference table to answer the question.
8) The preference table shows the results of an election among three candidates, A, B, and C.
Number of votes
First choice
Second choice
Third choice
7
A
B
C
6
B
C
A
3
B
A
C
(a) Using the plurality method, who is the winner?
(b) Is the majority criterion satisfied?
Rank the choices as requested.
9) An airline is deciding which type of movie to show during a flight. The passengers are asked to rank the
following four choices: (A)ction, (C)omedy, (M)usical, and (Drama). The results are summarized below.
Number of Ballots
Preference 10 13 19 8
1st
A M C M
2nd
M A M D
3rd
C D D C
4th
D C A A
4
C
A
D
M
11
D
M
A
C
Rank the four choices using the extended plurality method.
Solve the problem.
10) Eighteen voters are using the Borda method to select one of four candidates: a, b, c, or d. If Candidate a receives
35 Borda points, Candidate b receives 28 Borda points, and Candidate c receives 20 Borda points, how many
Borda points does Candidate d receive? Who wins the Borda election?
2
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
11) An election involving 6 candidates and 20 voters is held and the results of the election are to be determined
using the Borda count method. The minimum number of points a candidate can receive is
A) 120 points.
C) 60 points.
B) 15 points
D) 20 points
SHORT ANSWER. Answer each question. Circle your final answer. Show all work.
Determine the answer as requested.
12) The city council gave a questionnaire to its citizens, asking them to rank their priorities for next yearʹs budget.
People were asked to rank the following four choices: (P)olice, (R)oads, (S)chools, and (T)rash removal. The
results are summarized in the table below.
Number of Ballots
Preference 15 30 18 17
1st
P T P R
2nd
R R S T
3rd
S S T S
4th
T P R P
10
R
S
P
T
2
S
P
R
T
Which option is selected using the Borda count method?
Rank the choices as requested.
13) An airline is deciding which type of movie to show during a flight. The passengers are asked to rank the
following four choices: (A)ction, (C)omedy, (M)usical, and (Drama). The results are summarized below.
Number of Ballots
Preference 10 13 19 8
1st
A M C M
2nd
M A M D
3rd
C D D C
4th
D C A A
4
C
A
D
M
11
D
M
A
C
Rank the four choices using the extended Borda count method.
Determine the answer as requested.
14) The city council gave a questionnaire to its citizens, asking them to rank their priorities for next yearʹs budget.
People were asked to rank the following four choices: (P)olice, (R)oads, (S)chools, and (T)rash removal. The
results are summarized in the table below.
Number of Ballots
Preference 15 30 18 17
1st
P T P R
2nd
R R S T
3rd
S S T S
4th
T P R P
10
R
S
P
T
2
S
P
R
T
Which option is selected using the plurality-with-elimination method?
3
Use the preference table to answer the question.
15) The preference table shows the results of a straw vote among three candidates, A, B, and C.
Number of votes
First choice
Second choice
Third choice
7
A
B
C
6
B
C
A
3
C
B
A
(a) Using the plurality-with-elimination method, which candidate wins the straw vote?
(b) In the actual election, the 3 voters in the last column who voted C, B, and A, in that order, change their votes
to A, B, C. Using plurality-with-elimination method, which candidate wins the actual election.
(c) Is the monotonicity criterion satisfied?
Rank the choices as requested.
16) An airline is deciding which type of movie to show during a flight. The passengers are asked to rank the
following four choices: (A)ction, (C)omedy, (M)usical, and (Drama). The results are summarized below.
Number of Ballots
Preference 10 13 19 8
1st
A M C M
2nd
M A M D
3rd
C D D C
4th
D C A A
4
C
A
D
M
11
D
M
A
C
Rank the four choices using the extended plurality-with-elimination method.
Solve the problem.
17) Suppose 8 candidates a, b, c, d, e, f, g, h are involved in a Pairwise Comparison Method election. The number of
pairwise comparisons each candidate wins is shown in the table.
Candidate
a b c d e f g h
Number of Pairwise Comparisons Won 4 5 ? 3 4 2 1 4
How many comparisons does candidate c win?
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
18) An election is held among six candidates (A, B, C, D, E, and F). Using the method of pairwise comparisons, A
1
gets 5 points; B gets 4 points; C gets 2 points; D gets 1 points, and E gets 0 points. How many points does F
2
get?
A) 3
B) 2
C) 2
4
1
2
D) 1
1
2
SHORT ANSWER. Answer each question. Circle your final answer. Show all work.
Determine the answer as requested.
19) The city council gave a questionnaire to its citizens, asking them to rank their priorities for next yearʹs budget.
People were asked to rank the following four choices: (P)olice, (R)oads, (S)chools, and (T)rash removal. The
results are summarized in the table below.
Number of Ballots
Preference 15 30 18 17
1st
P T P R
2nd
R R S T
3rd
S S T S
4th
T P R P
10
R
S
P
T
2
S
P
R
T
Which option is selected using the pairwise comparison method?
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
For an election with four candidates (A, B, C, and D) we have the following preference schedule:
Number of Voters
1st choice
2nd choice
3rd choice
4th choice
6 3 5 8
D D A C
B A C A
A B B D
C C D B
20) The ranking of the candidates using the extended pairwise comparisons method is
A) first: A; second: D; third: C; fourth: B.
B) first: D; second: A; third: C; fourth: B.
C) first: D; second: C; third: A; fourth: B.
D) first: A; second: C; third: D; fourth: B.
21) In this election,
A) B is a Condorcet candidate.
B) A is a Condorcet candidate.
C) every candidate is a Condorcet candidate.
D) there is no Condorcet candidate.
For an election with candidates A, B and C, we have the following preference schedule:
Number of voters 27 24
1st choice
A B
2nd choice
C C
3rd choice
B A
2
C
B
A
22) Which fairness criterion is violated in this election using the Borda count method if candidate B drops out of
the election and a reelection is held?
A) independence-of-irrelevant-alternatives criterion
B) monotonicity criterion
C) majority criterion
D) Condorcet criterion
5
Solve the problem.
23) ʺIf choice X is a winner of an election and, in a reelection, the only changes in the ballots are changes that only
favor X, then X should remain a winner of the election.ʺ This fairness criterion is called the
A) Condorcet criterion.
B) majority criterion.
C) monotonicity criterion.
D) independence of irrelevant alternatives criterion.
24) An election is held among four candidates (A, B, C, and D). Using a voting method we will call X, the winner of
the election is candidate A. However, candidate D has more than 50% of the possible votes. Based on this
information, we can say that voting method X violates the
A) independence of irrelevant alternatives criterion.
B) Condorcet criterion.
C) majority criterion.
D) monotonicity criterion.
25) An election is held for president of the United States. Three candidates are running: a Democrat, a Republican,
and an Independent. A certain voter prefers the Independent candidate over the other two, but realizing
(because of all the pre-election polls) the race is going to be a close race between the Democrat and the
Republican and that the Independent doesnʹt have a chance, he votes instead for his second choice (his
preference between the Democrat and the Republican). This is an example of
A) insincere voting.
B) the majority criterion.
C) the monotonicity criterion.
D) the independence of irrelevant alternatives criterion.
Answer the question.
26) Choose the sentence or sentences that accurately restate Arrowʹs Impossibility Theorem.
I. It is mathematically impossible for any democratic voting system to satisfy any of the four fairness criteria.
II. It is mathematically impossible for any democratic voting system to satisfy all of the four fairness criteria.
III. It is mathematically impossible for any democratic voting system to satisfy some of the four fairness criteria.
IV. It is mathematically impossible for any democratic voting system to satisfy any more than one of the four
fairness criteria.
A) II only
B) IV only
C) I, III, and IV
6
D) I, II, and III
Answer Key
Testname: CH 1 REVIEW
1)
2)
3)
4)
No
C
D
B
Number of votes
First choice
5) Second choice
Third choice
Fourth choice
5 4 2
D D A
A B C
B A B
C C D
1
C
A
B
D
6) 40,320
7) P
8) B; yes
9) C, M, D, A
10) 97; d
11) D
12) R
13) M, C, D, A
14) T
15) B; A; no
16) M, C, D, A
17) 5
18) C
19) T
20) D
21) B
22) A
23) C
24) C
25) A
26) A
7