Sample test problems – Chapter 1: Voting methods
The next three questions refer to an election with 4 candidates (A, B, C, and D), 71 voters
and preference schedule given by the following table.
1.
Using the plurality method the winner of the election is _______
2.
Using the method of pairwise comparisons the winner of the election is _______
3.
Using the Borda count method, which candidate comes in last? _______
4.
How many people voted in the election below? _______
5.
What is the total number of pairwise comparisons in an election among 6
candidates? _______
6.
"If there is a choice that has a majority of the first-place votes in an election, then
that choice should be the winner of the election." This fairness criterion is called the
A
B
C
D
E
monotonicity criterion
Condorcet criterion
majority criterion
independence of irrelevant alternatives criterion
None of the above
7.
"If in an election there is a Condorcet candidate, then such a candidate should be
the winner of the election." This statement is another way to phrase the
A
independence of irrelevant alternatives criterion
Math 110 test Chapters 1 and 3
1
B
C
D
E
Condorcet criterion
monotonicity criterion
majority criterion
None of the above
8.
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. Due to an irregularity in
the original vote count a recount is required. Before the recount takes place, candidate B
drops out of the race. In the recount, still using voting method X, candidate D wins the
election. Based on this information, we can say that voting method X violates the
A
B
C
D
E
majority criterion
monotonicity criterion
Condorcet criterion
independence of irrelevant alternatives criterion
None of the above
9.
An election is held among six candidates (A, B, C, D, E, and F) Using the method
of pairwise comparisons A gets 6 points; B gets 3 points; C gets 2.5 points; D gets 1.5
points, and E gets 1 point. How many points does F get?
A
B
C
D
E
F
1
1.5
2
2.5
3
none of the above
10.
An election involving 5 candidates and 30 voters is held and the results of the
election are to be determined using the Borda count method. The maximum number of
points a candidate can receive is
A
B
C
D
E
30
50
90
150
none of the above
11.
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 doesn't 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
Math 110 test Chapters 1 and 3
2
B
C
D
E
the independence of irrelevant alternatives criterion
stupidity
the monotonicity criterion
None of the above
12.
The method of pairwise comparisons violates
A
B
C
D
E
the majority criterion
none of the four criteria
the independence of irrelevant alternatives criterion
the Condorcet criterion
None of the above
The next 5 questions refer to an election with four
candidates (A, B, C, and D), and with the
following preference schedule:
13.
How many people voted in this election?
________
Number of
voters
9 7 4 2
1st choice
A D B C
2nd choice
B C D B
3rd choice
C B C D
4th choice
D A A A
14.
Using the plurality method, which candidate wins the election? ________
15.
Using the Borda count method, which candidate wins the election? _______
16.
Using the Instant Runoff (plurality-with-elimination) method, which candidate
wins the election? ________
17.
Using the Borda count method, which candidate wins the election if candidate B
drops out of the race? ________
18.
In a round robin tournament, every player plays against every other player. If 8
players are entered in a round robin golf tournament, how many matches will be
played? ________
19.
“If 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.”
This fairness criterion is called the
A
B
C
D
E
Condorcet criterion.
monotonicity criterion.
independence of irrelevant alternatives criterion.
majority criterion.
None of the above
Math 110 test Chapters 1 and 3
3
20.
An election is held among 5 candidates (A, B, C, D and E) using the Borda count
method. There are 30 voters. If candidate A received 93 points, candidate B received 87
points, candidate C received 48 points, and candidates D and E tied, how many points did
candidates D and E each receive?
A
B
C
D
E
111
222
228
Cannot be determined with this information
None of the above
21.
An election is held among seven candidates (A, B, C, D, E, F and G). There are
7,000 voters. Using the method of pairwise comparisons, A, B, and C win 3 pairwise
comparisons each. D, E, and F each win 2 pairwise comparisons, and G wins all the rest.
In this election
A
B
C
D
E
A, B, and C are Condorcet candidates.
there is no Condorcet candidate.
G is a Condorcet candidate.
there is not enough information to determine if there is a Condorcet candidate.
None of the above
22.
An election involving 7 candidates and 20 voters is held and the results of the
election are to be determined using the Borda count method. Assuming there isn't a tie for
first place, the minimum number of points a winning candidate can receive is
A
B
C
D
E
23.
81 points.
71 points.
91 points.
61 points.
None of the above
Arrow’s Impossibility Theorem implies
A that in every election, no matter what voting method we use, at least one of the four
fairness criteria will be violated.
B that every voting method can potentially violate each one of the four fairness
criteria.
C that in every election, each of the voting methods must produce a different winner.
D that it is impossible to have a voting method that satisfies all four of the fairness
criteria.
Sample test problems – Chapter 2: Fair Division
Math 110 test Chapters 1 and 3
4
The next two questions refer to the following situation: Three players (one divider
and two choosers) are going to divide a cake fairly using the lone divider method.
The divider cuts the cake into three slices (s 1 , s 2 , and s 3 ).
24.
If the choosers declarations are Chooser 1: {s2 } and Chooser 2: {s3 }, which of the
following is a fair division of the cake?
A
B
C
D
E
25.
Chooser 1 gets s3 ; Chooser 2 gets s2 ; Divider gets s1 .
Chooser 1 gets s1 ; Chooser 2 gets s2 ; Divider gets s3 .
Chooser 1 gets s2 ; Chooser 2 gets s3 ; Divider gets s1 .
Chooser 1 gets s2 ; Chooser 2 gets s1 ; Divider gets s3 .
None of the above
Suppose the choosers value the slices as follows:
Which of the following is a fair division of the cake?
A
B
C
D
E
Chooser 1 gets s2 ; Chooser 2 gets s1 ; Divider gets s3 .
Chooser 1 gets s3 ; Chooser 2 gets s2 ; Divider gets s1 .
Chooser 1 gets s2 ; Chooser 2 gets s3 ; Divider gets s1 .
Chooser 1 gets s1 ; Chooser 2 gets s2 ; Divider gets s3 .
None of the above
The next four questions refer to the following example: Four heirs (A, B, C, and D)
must divide fairly an estate consisting of three items - a house, a cabin and a boat using the method of sealed bids. The players' bids (in dollars) are:
26.
The original fair share of player B is worth
A
B
C
D
E
$64,000
$62,000
$62,500
$65,500
None of the above
Math 110 test Chapters 1 and 3
5
27.
In the initial allocation, player B
A
B
C
D
E
28.
After the initial allocation to each player is made there is a surplus of
A
B
C
D
E
29.
gets the cabin and an additional $15,500 from the estate.
gets $65,500 cash
gets the house and pays the estate $134,500.
gets the boat and an additional $53,500 from the estate.
None of the above
$0
$6,000
$12,000
$24,000
None of the above
After all is said and done, the final allocation to player B is:
A
B
C
D
E
the house minus $134,500 in cash.
$65,500 in cash.
the house minus $128,500 in cash
the house plus $6,000 in cash.
None of the above
The next six questions refer to the following: Four players (A, B, C, and D) agree to
divide the 15 items shown below by lining them up in order and using the method of
markers. The player's bids are as indicated.
30.
A
B
C
D
E
Item 3
goes to A
goes to B
goes to C
goes to D
is left over
30.
A
B
C
D
E
Item 5
goes to A
goes to B
goes to C
goes to D
is left over
30.
A
B
C
D
E
Item 7
goes to A
goes to B
goes to C
goes to D
is left over
31.
A
B
C
D
E
Item 10
goes to A
goes to B
goes to C
goes to D
is left over
31.
A
B
C
D
E
Item 13
goes to A
goes to B
goes to C
goes to D
is left over
31.
A
B
C
D
E
Item 15
goes to A
goes to B
goes to C
goes to D
is left over
Math 110 test Chapters 1 and 3
6
32.
An estate consisting of a car, a boat, a house and a collection of rare books must
be divided fairly among five heirs. This type of problem is called
A
B
C
D
E
a discrete fair division problem
a mixed fair division problem
a continuous fair division problem
the method of sealed bids
None of the above
33.
Joe and Bill want to divide a cake using the divider-chooser method. They draw
straws, and it is determined that Bill will be the divider and Joe the chooser. Assuming
that each plays the game correctly, which of the following statements cannot be true?
(Select "None of the above" if the other four statements are true.)
A
B
C
D
E
Joe believes that his share is worth 60% of the cake; Bill believes that his
share is worth 50% of the cake.
Bill believes that Joe's share is worth 50% of the cake; Joe believes that his
share is worth 60% of the cake.
Joe believes that his share is worth 50% of the cake; Bill believes that his
share is worth 50% of the cake.
Bill believes that his share is worth 60% of the cake; Joe believes that his
share is worth 50% of the cake.
None of the above
34.
Sue and Tom are getting a divorce. Except for the house they own very little of
value so they agree to divide the house fairly using the method of sealed bids. Sue bids
100,000 and Tom bids 90,000. After all is said and done, the final outcome is
A
B
C
D
E
Sue gets the house and pays Tom 50,000
Sue gets the house and pays Tom 47,500
Sue gets the house and pays Tom 45,000
Sue gets the house and pays Tom 55,000
None of the above
The next three questions refer to the following situation:
Aaron and Brittney want to divide fairly the chocolatebanana cake shown in the figure using the dividerchooser method. The total cost of the cake was $6.00.
Aaron values chocolate five times as much as he values
banana, while Brittney values banana twice as much as
she values chocolate.
Math 110 test Chapters 1 and 3
7
35.
In Aaron’s eyes, the piece shown in the figure is worth
A
B
C
D
E
$2.00
$3.00
$2.50
$3.50
None of the above
36.
In Aaron’s eyes, the piece shown in Figure 3.3 is
worth
(the angles are 45 degrees and 90 degrees)
A
B
C
D
E
$1.50
$1.75
$1.25
$1.00
None of the above
37.
In Brittney’s eyes, the piece shown in the figure is worth
(the angles are 45 degrees and 90 degrees)
A
B
C
D
E
$2.00
$2.50
$3.00
$3.50
none of the above
38.
Three players (one divider and two choosers) are going to divide a cake fairly
using the lone divider method. The divider cuts the cake into three slices (s1 , s2 , and s3 ). If
the choosers declarations are Chooser 1: {s1 , s3 } and Chooser 2: {s3 }, which of the
following is a fair division of the cake?
A
B
C
D
E
Chooser 1 gets s2 ; Chooser 2 gets s1 ; Divider gets s3 .
Chooser 1 gets s3 ; Chooser 2 gets s1 ; Divider gets s2 .
Chooser 1 gets s1 ; Chooser 2 gets s3 ; Divider gets s2 .
Chooser 1 gets s1 ; Chooser 2 gets s2 ; Divider gets s3 .
None of the above
39.
Three players (one divider and two choosers)
are going to divide a cake fairly using the lone divider
method. The divider cuts the cake into three slices (s1 ,
s2 , and s3 ). Suppose the choosers value the slices as
follows:
s1
s2
s3
Chooser 1 38% 32% 30%
Chooser 2 29% 29% 41%
Math 110 test Chapters 1 and 3
8
A
B
C
D
E
Chooser 1 gets s2 ; Chooser 2 gets s3 ; Divider gets s1 .
Chooser 1 gets s2 ; Chooser 2 gets s1 ; Divider gets s3 .
Chooser 1 gets s1 ; Chooser 2 gets s3 ; Divider gets s2 .
Chooser 1 gets s3 ; Chooser 2 gets s2 ; Divider gets s1 .
None of the above
Questions 6 through 9 refer to the following example: Four heirs (A, B, C, and D)
must divide fairly an estate consisting of three items – a house, a parcel of land, and
a diamond ring – using the method of sealed bids. The players’ bids (in dollars) are:
A
B
C
D
House 140,000 155,000 160,000 148,000
40.
100,000 75,000
Ring
40,000
35,000
36,000
25,000
$260,000.
$65,000.
$86,667.
$130,000.
None of the above
gets the ring and pays the estate $25,000.
gets the house and pays the estate $75,000.
gets the land and pays the estate $15,000.
gets the ring and an additional $25,000 from the estate.
None of the above
After the initial allocation to each player is made there is a surplus of
A
B
C
D
E
43.
90,000
In the initial allocation, player A
A
B
C
D
E
42.
80,000
The original fair share of player A is worth
A
B
C
D
E
41.
Land
$186,000.
$0.
$101,000.
$29,000.
None of the above
After all is said and done, the final allocation to player B is:
A
$99,000 in cash.
Math 110 test Chapters 1 and 3
9
B
C
D
E
$77,250 in cash.
the ring and $32,250 in cash.
$70,000 in cash.
None of the above
Questions 10 through 12 refer to the following: Four players (A, B, C, D) agree to
divide the 12 items shown in Figure 3.5 using the method of markers. The player’s
bids are as indicated.
44.
A
B
C
D
E
Item 4
goes to A
goes to B
goes to C
goes to D
is left over
44.
Item 10
44.
A
B
C
D
E
goes to A
goes to B
goes to C
goes to D
is left over
A
B
C
D
E
Item 7
goes to A
goes to B
goes to C
goes to D
is left over
45.
Which player receives the largest number of items? _________
46.
Which of the following is a discrete fair division problem?
A
B
C
D
E
dividing a chocolate cake
dividing a gallon of milk
dividing a liberated country
dividing an collection of jewels
None of the above
47.
Phil and Rick are dividing a business. Except for the factory, they own very little
of value so they agree to divide the factory fairly using the method of sealed bids. Phil
bids $200,000 and Rick bids $190,000. After all is said and done, the final outcome is
A
B
C
D
E
Phil gets the factory and pays Rick $97,500.
Phil gets the factory and pays Rick $100,000.
Phil gets the factory and pays Rick $105,000.
Phil gets the factory and pays Rick $95,000.
None of the above
Math 110 test Chapters 1 and 3
10