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16. Under majority rule, the order in which items are voted on is
a. unimportant, and this is a lesson of the Condorcet paradox.
b. unimportant, and this is a lesson of Arrow’s impossibility theorem.
c. important, and this is a lesson of the Condorcet paradox.
d. important, and this is a lesson of Arrow’s impossibility theorem.
17. The Condorcet paradox shows that
a. allocations of resources based on majority rule are always inefficient.
b. problems in counting votes can negate legitimate democratic outcomes.
c. the order on which things are voted can affect the result.
d. transitive preferences are inconsistent with rationality.
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18. Which of the following is a lesson from the Condorcet paradox?
a. If voters are choosing a point along a line, then majority rule will pick the most preferred point
of the median voter.
b. Under certain conditions, there is no scheme for aggregating individual preferences.
c. When there are more than two options, deciding the order in which to vote can have a powerful
influence over the outcome of an election.
d. Majority voting always indicates what outcome a society really wants.
19. One implication of the Condorcet paradox is
a. that the order in which things are voted on can affect the result.
b. that the order in which things are voted on is irrelevant.
c. that you do not want to be in charge of arranging which items are voted upon first.
d. that when there are only two items being voted on the order matters.
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20. Which of the following is not correct?
a. Pairwise voting never produces transitive preferences.
b. The order of pairwise voting can affect the result.
c. Majority voting by itself does not tell us what outcome a society really wants.
d. No voting system can satisfy all of the following properties: unanimity, transitivity,
independence of irrelevant alternatives, and no dictators.
21. As an alternative to pairwise majority voting, each voter could be asked to rank the possible
outcomes, giving 1 point to her lowest choice, 2 points to her second–lowest choice, 3 points to
her third-lowest choice, and so on. This voting method is called a(n)
a. median vote.
b. pairwise minority vote.
c. Borda count.
d. Arrow count.
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22. Suppose that residents of a town are asked to vote on the best day to improve the safety of an
intersection. The three choices are: a stoplight, a 4-way stop, and a 2-way stop. The mayor asks
the residents to assign 3 points to their first choice, 2 points to their second choice, and 1 point to
their last choice. The intersection will be controlled by the method that receives the most points.
This voting scheme is called
a. Arrow’s impossibility theorem.
b. the Condorcet paradox.
c. a Borda count.
d. the median voter theorem.
23. Suppose that residents of a town are asked to vote on the best way to improve the safety of an
intersection. The three choices are: a stoplight, a 4-way stop, and a 2-way stop. When the mayor
asks the residents to choose between a stoplight and a 4-way stop, the residents choose a 4-way
stop. Then, when the mayor asks them to choose between a 4-way stop and a 2-way stop, they
choose a 2-way stop. However, if the mayor firsts asks the residents to choose between a 4-way
stop and a 2-way stop, they choose a 2-way stop. Then, when the mayor asks the residents to
choose between a 2-way stop and a stoplight, they choose a stoplight. What does this example
illustrate?
a. Arrow’s impossibility theorem
b. the Condorcet paradox
c. a Borda count
d. the median voter theorem
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Table 22-1
Three friends — Linda, Stephanie, and Jamie — are deciding where to go together for vacation.
They all agree that they should go to one of three places: France, Greece, or Italy. They also
agree that they will have two pairwise votes to determine where to go on vacation, with the
majority determining the outcome on each vote. The first, second, and third choices for each
person are as indicated in the table below.
Linda
Stephanie
Jamie
First choice
France
Greece
Italy
Second choice
Greece
Italy
France
Third choice
Italy
France
Greece
24. Refer to Table 22-1. If the first vote pits France against Greece and the second vote pits Italy
against the winner of the first vote, then the outcome is as follows:
a. France wins the first vote and Italy wins the second vote, so they go to Italy.
b. France wins the first vote and France wins the second vote, so they go to France.
c. Greece wins the first vote and Greece wins the second vote, so they go to Greece.
d. Greece wins the first vote and Italy wins the second vote, so they go to Italy.
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25. Refer to Table 22-1. If the first vote pits France against Italy and the second vote pits Greece
against the winner of the first vote, then the outcome is as follows:
a. France wins the first vote and Greece wins the second vote, so they go to Greece.
b. France wins the first vote and France wins the second vote, so they go to France.
c. Italy wins the first vote and Italy wins the second vote, so they go to Italy.
d. Italy wins the first vote and Greece wins the second vote, so they go to Greece.
26. Refer to Table 22-1. If the first vote pits Greece against Italy and the second vote pits France
against the winner of the first vote, then the outcome is as follows:
a. Greece wins the first vote and France wins the second vote, so they go to France.
b. Greece wins the first vote and Greece wins the second vote, so they go to Greece.
c. Italy wins the first vote and Italy wins the second vote, so they go to Italy.
d. Italy wins the first vote and France wins the second vote, so they go to France.
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27. Refer to Table 22-1. Depending on the order of the pairwise voting,
a. the friends could go to either France, Italy, or Greece.
b. the friends could go to either France or Italy, but they will not go to Greece.
c. the friends could go to either Italy or Greece, but they will not go to France.
d. the friends could go to either France or Greece, but they will not go to Italy.
28. Refer to Table 22-1. If the friends change their minds and decide to choose a vacation
destination using a Borda count, then
a. the friends will go to France.
b. the friends will go to Greece.
c. the friends will go to Italy.
d. A Borda count will not result in a single winner in this case.
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Table 22-2
Three longtime friends–Allen, Brian, and Cody–are deciding how they will spend their Sunday
afternoon. They all agree that they should do one of three things: go to a movie, play golf, or go to
a baseball game. They also agree that they will have two pairwise votes to determine how to spend
their afternoon, with the majority determining the outcome on each vote. The first, second, and
third choices for each person are as indicated in the table below.
Allen
Brian
First choice
Baseball Game
Golf
Second choice
Golf
Movie
Third choice
Movie
Baseball Game
29. Refer to Table 22-2. If (1) the first vote pits “baseball game” against “movie,” and (2) the second
vote pits “golf” against the winner of the first vote, then the outcome is as follows:
a. “Baseball game” wins the first vote and “baseball game” wins the second vote, so they go to a
baseball game.
b. “Baseball game” wins the first vote and “golf” wins the second vote, so they go to the golf.
c. “Movie” wins the first vote and “movie” wins the second vote, so they go to a movie.
d. “Movie” wins the first vote and “golf” wins the second vote, so they play golf.
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30. Refer to Table 22-2. If (1) the first vote pits “baseball game” against “golf,” and (2) the second
vote pits “movie” against the winner of the first vote, then
a. “Baseball game” wins the first vote and “baseball game” wins the second vote, so they go to a
baseball game.
b. “Baseball game” wins the first vote and “movie” wins the second vote, so they go to a movie.
c. “golf” wins the first vote and “golf” wins the second vote, so they play golf.
d. “golf” wins the first vote and “movie” wins the second vote, so they go to a movie.
31. Refer to Table 22-2. Which of the following statements is correct?
a. In a pairwise election, “movie” beats “golf.”
b. In a pairwise election, “golf” beats “baseball game.”
c. In a pairwise election, “baseball game” beats “movie.”
d. None of the above is correct.
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32. Refer to Table 22-2. Which of the following statements is correct?
a. In a pairwise election, “golf“ beats “movie.“
b. In a pairwise election, “baseball game” beats “golf.“
c. In a pairwise election, “movie” beats “baseball game.”
d. All of the above are correct.
33. Refer to Table 22-2. Which of the following statements is correct regarding the Condorcet
paradox and the results of pairwise voting by Allen, Brian, and Cody?
a. The paradox implies that pairwise voting never produces transitive preferences, and so the
voting by Allen, Brian, and Cody fails to produce transitive preferences.
b. The paradox implies that pairwise voting sometimes (but not always) produces transitive
preferences, and the voting by Allen, Brian, and Cody does produce transitive preferences.
c. The paradox implies that pairwise voting sometimes (but not always) fails to produce transitive
preferences, and the voting by Allen, Brian, and Cody fails to produce transitive preferences.
d. The paradox does not apply to the case at hand, because Brian’s preferences are not
individually transitive.
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34. Refer to Table 22-2. If Allen, Brian, and Cody use a Borda count, rather than pairwise majority
voting, to decide how to spend their afternoon, then they will
a. watch a movie.
b. play golf.
c. watch a baseball game.
d. None of the above is correct; a Borda count fails to produce a winner in this instance.
Table 22-3
Three family members — Seamus, Maeve, and Siobhan — are deciding what type of movie to
attend. The three choices are an action adventure, comedy, or horror. The first, second, and third
choices for each person are as indicated in the table below.
Seamus
Maeve
Siobhan
First Choice
Comedy
Action
Horror
Second Choice
Horror
Horror
Comedy
Third Choice
Action
Comedy
Action
35. Refer to Table 22-3. If the voting method is a Borda count, which alternative will be chosen?
a. Comedy
b. Action
c. Horror
d. None of the above is correct; a Borda count fails to produce a winner in this instance.
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36. Refer to Table 22-3. Suppose the three decide to make the decision based on pairwise majority
voting. If they first choose between Action and Comedy and then choose between the winner of
the first vote and Horror, which movie alternative will win?
a. Action
b. Comedy
c. Horror
d. There is no clear winner – Comedy and Horror will tie.
37. Refer to Table 22-3. Suppose the three decide to make the decision based on pairwise majority
voting. If they first choose between Action and Horror and then choose between the winner of
the first vote and Comedy, which movie alternative will win?
a. Action
b. Comedy
c. Horror
d. There is no clear winner – Action and Horror will tie.
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Table 22-4
The fortunate residents of Anytown have a budget surplus. The mayor decided that it is only fair
to have the residents vote on what to do with the surplus. The mayor has narrowed the options
down to three possible projects: a playground, a library, or a swimming pool. The voters fall into
three categories and have preferences as illustrated in the table.
Voter Types
Residents with
Young Children
Residents with
Older Children
Residents with No
Children
Percent of Electorate
45
35
20
First Choice
Playground
Swimming Pool
Library
Second Choice
Library
Playground
Swimming Pool
Third Choice
Swimming Pool
Library
Playground
38. Refer to Table 22-4. If the mayor asks the residents to choose between the playground and the
library using pairwise voting,
a. the playground wins by 45%.
b. the playground wins by 60%.
c. the library wins by 20%.
d. the library wins by 80%.
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39. Refer to Table 22-4. If the mayor asks the residents to choose between the library and the
swimming pool using pairwise voting,
a. the library wins by 30%.
b. the library wins by 65%.
c. the swimming pool wins by 10%.
d. the swimming pool wins by 35%.
40. Refer to Table 22-4. If the mayor asks the residents to choose between the playground and the
swimming pool using pairwise voting,
a. the playground wins by 10%.
b. the playground wins by 45%.
c. the swimming pool wins by 10%.
d. the swimming pool wins by 55%.
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41. Refer to Table 22-4. Which of the following statements is correct regarding the results of
pairwise voting in Anytown?
a. The results of pairwise voting depend on the order of the pairs but satisfy the transitivity
property.
b. The results of pairwise voting do not depend on the order of the pairs and satisfy the
transitivity property.
c. The results of pairwise voting depend on the order of the pairs and do not satisfy the
transitivity property.
d. The results of pairwise voting do not depend on the order of the pairs and do not satisfy the
transitivity property.
42. Refer to Table 22-4. If the mayor decides to use a Borda count rather than pairwise voting,
a. the swimming pool will win.
b. the library will win.
c. the playground will win.
d. the results will be the same as with pairwise voting.
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Table 22-5
The citizens of Anytown will decide whether to build a new library, a recreation center, or an
arena. Exactly one of the three choices will prevail, and the choice will be made by way of
pairwise voting, with the majority determining the outcome on each vote. The preferences of the
voters are summarized in the table below.
Voter Type
Type 1
Type 2
Type 3
Percent of Electorate
40
35
25
First choice
Library
Recreation Center
Arena
Second choice
Recreation Center
Arena
Library
Third choice
Arena
Library
Recreation Center
43. Refer to Table 22-5. If (1) the first vote pits “library” against “recreation center,” and (2) the
second vote pits “arena” against the winner of the first vote, then the outcome is as follows:
a. “Library” wins the first vote and “library” wins the second vote, so they build a library.
b. “Library” wins the first vote and “arena” wins the second vote, so they build an arena.
c. “recreation center” wins the first vote and “recreation center” wins the second vote, so they
build a recreation center.
d. “recreation center” wins the first vote and “arena” wins the second vote, so they build an arena.
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44. Refer to Table 22-5. If (1) the first vote pits “library” against “arena,” and (2) the second vote
pits “recreation center” against the winner of the first vote, then the outcome is as follows:
a. “Library” wins the first vote and “library” wins the second vote, so they build a library.
b. “Library” wins the first vote and “recreation center” wins the second vote, so they build a
recreation center.
c. “arena” wins the first vote and “arena” wins the second vote, so they build an arena.
d. “arena” wins the first vote and “recreation center” wins the second vote, so they build a
recreation center.
45. Refer to Table 22-5. Which of the following statements is correct?
a. In a pairwise election, “library” beats “arena.”
b. In a pairwise election, “arena” beats “recreation center.”
c. In a pairwise election, “library” beats “recreation center.”
d. All of the above are correct.
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46. Refer to Table 22-5. Which of the following statements is correct?
a. In a pairwise election, “arena” beats “library.”
b. In a pairwise election, “library” beats “recreation center.”
c. In a pairwise election, “recreation center” beats “arena.”
d. All of the above are correct.
47. Refer to Table 22-5. Which of the following statements is correct regarding the Condorcet
paradox and the results of pairwise voting in Anytown?
a. The results of pairwise voting depend on the order of the pairs and preferences are transitive.
b. The results of pairwise voting depend on the order of the pairs, but preferences are not
transitive.
c. The results of pairwise voting do not depend on the order of the pairs, but preferences are
transitive.
d. The results of pairwise voting do not depend on the order of the pairs and preferences are not
transitive.
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48. Refer to Table 22-5. If the citizens of Anytown use a Borda count, rather than pairwise majority
voting, to decide what to build, then they will build a new
a. library.
b. recreation center.
c. arena.
d. None of the above is correct; a Borda count fails to produce a winner in this instance.
Table 22-6
Voter Type
Type 1
Type 2
Type 3
# Voters
40
15
45
First choice
C
B
A
Second choice
B
A
C
Third choice
A
C
B
49. Refer to Table 22–6. The table shows the preferences of 100 voters over three possible
outcomes: A, B, and C. If a Borda count election were held among these voters, giving three
points to each voter’s first choice, two points to the second choice, and one point to the last
choice, which outcome would win the election?
a. Outcome A
b. Outcome B
c. Outcome C
d. Either outcome A or outcome C since these have the same total score.
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50. Refer to Table 22-6. The table shows the preferences of 100 voters over three possible
outcomes: A, B, and C. Which of the following statements is true?
a. In pairwise majority voting, B is preferred to A, A is preferred to C, and B is preferred to C.
b. In pairwise majority voting, C is preferred to B, B is preferred to A, and C is preferred to A.
c. In pairwise majority voting, B is preferred to A, A is preferred to C, and C is preferred to B.
d. In pairwise majority voting, A is preferred to C, C is preferred to B, and A is preferred to B.
51. Refer to Table 22-6. The table shows the preferences of 100 voters over three possible
outcomes: A, B, and C. In pairwise majority voting in which voters choose first between A and B
and then choose between the winner of the first vote and C,
a. outcome A will win the election.
b. outcome B will win the election.
c. outcome C will win the election.
d. the outcome of the election cannot be determined with the given information.