Quarter 2 Review
1. Arrows
Impossibility Theorem
• For any voting system it is possible
to find a set of voters' preferences
that will cause the voting system to
violate a condition deemed desirable
for a fair voting system. These
conditions may include the
Condorcet Winner Criterion and the
Independence of Irrelevant
Alternatives condition.
2. Explain the Condorcet
Winner Criterion
• A voting system satisfies the
Condorcet Winner Criterion if the
winner of an election is also the
Condorcet winner, if a Condorcet
winner exists.
3. Use the preference schedule of 23 voters shown below
to answer the following question(s).
Number of Voters
8
5
First choice
A
C
Second choice C
A
Third choice
B
B
6
B
C
A
4
B
A
C
Which candidate, if any, wins in a
majority rule election?
There is no majority-rule winner.
4. Use the preference schedule of 23 voters shown below
to answer the following question(s).
Number of Voters
8
5
First choice
A
C
Second choice C
A
Third choice
B
B
6
B
C
A
4
B
A
C
If a rank method is used, which
candidate, if any, wins in a straight
plurality election?
B wins in a straight plurality vote.
5. Use the preference schedule of 23 voters shown below
to answer the following question(s).
Number of Voters
8
5
First choice
A
C
Second choice C
A
Third choice
B
B
6
B
C
A
4
B
A
C
Which candidate, if any, wins if an
election is held between A and C
and the winner of that race runs
against B? Who wins the final
election?
A wins.
Use the preference schedule of 23 voters shown below to
answer the following question(s).
First choice
Second choice
Third choice
8
A
C
B
Number of Voters
5
6
C
B
A
C
B
A
4
B
A
C
If a Borda count is used that assigns 3
points for a first place vote, 2 points for
a second place vote, and 1 point for a
third place vote, who wins the election?
A wins
Use the preference schedule of 23 voters shown below to
answer the following question(s).
First choice
Second choice
Third choice
8
A
C
B
Number of Voters
5
6
C
B
A
C
B
A
4
B
A
C
Can the four voters in the last column
vote strategically to change the
outcome of question 6 to one they
would like better? Why or why not?
No. They cannot make B win the election and if
they switch any rankings to place C higher, then C
would win and this is their least desirable
outcome.
Seventeen board members vote on four candidates,
A, B, C, or D, for a new position on their board.
Their preference schedules are shown below.
First choice
Second choice
Third choice
Fourth choice
Number of Members
7
6
4
A
D
C
B
A
B
C
B
D
D
C
A
• Which candidate will be selected if
they use the Hare system?
D wins.
Seventeen board members vote on four candidates,
A, B, C, or D, for a new position on their board.
Their preference schedules are shown below.
First choice
Second choice
Third choice
Fourth choice
Number of Members
7
6
4
A
D
C
B
A
B
C
B
D
D
C
A
• Using the Hare system according to the
preference schedules shown, what
happens if A rejects the offer before the
ranking?
B wins.
Consider the following preference table:
Number of voters
3
2
2
First choice
A
B
C
Second choice B
C
B
Third choice
C
A
A
Which candidate will be chosen if majority
rule is used?
No one wins.
11. How many ways can one
rank 3 candidates if ties are
allowed?
• 3!+3C2+3C2+3C3=13
After their star pitcher moved to another town, the eight remaining
members of the company baseball team needed to select a new
pitcher. They used approval voting on the four prospects, and the
results are listed below. An “X” indicates an approval vote.
Alan
X
Bob
X
X
X
X
Chuck
X
David
X
X
X
X
X
Which pitcher is chosen if just one is to be selected?
Bob
X
X
X
X
X
What is agenda
manipulation?
The ability to control the winner of
an election by the selection of a
particular agenda.
Explain the difference between
sincere and strategic voting.
Sincere voting means submitting a
ballot that reflects the voter's true
preferences. Strategic voting means
submitting a ballot that does not
reflect the voter's true preferences but
will lead to an outcome the voter likes
better than would occur if the voter
voted sincerely.
15. If a voting system has three
or more alternatives, satisfies the
Pareto condition, always
produces a unique winner, and is
not a dictatorship, what
conclusion follows from the GS
theorem?
The voting method can be
manipulated.
Are there voting methods that are
group manipulable though not
individual manipulable?
Yes, plurality, for example.
A seventeen-member committee must elect one of
four candidates: R, S, T, or W. See the preference
schedule below.
First choice
Second choice
Third choice
Fourth choice
Number of Members
6
4
3
4
R
S
T
W
S
R
S
T
T
T
R
S
W
W
W
R
R wins using the plurality method. Could those
members who most prefer W vote strategically in
some way to change the outcome in a way that
will benefit them?
Yes. If these four voters ranked candidate T first,
T would win with 7 votes. Since the voters prefer
T to R, this outcome would be more desirable.
A seventeen-member committee must elect one of
four candidates: R, S, T, or W. See the preference
schedule below.
First choice
Second choice
Third choice
Fourth choice
Number of Members
6
4
3
4
R
S
T
W
S
R
S
T
T
T
R
S
W
W
W
R
R wins using the plurality method. Could those
members who most prefer T vote strategically in
some way to change the outcome in a way that
will benefit them?
Yes. If these three voters ranked candidate S
first, S would win with 7 votes. Since the
voters prefer S to R, this outcome would be
more desirable.
A seventeen-member committee must elect one of
four candidates: R, S, T, or W. See the preference
schedule below.
First choice
Second choice
Third choice
Fourth choice
Number of Members
6
4
3
4
R
S
T
W
S
R
S
T
T
T
R
S
W
W
W
R
R wins using the plurality method. Could those
members who most prefer S vote strategically in
some way to change the outcome in a way that
will benefit them?
No
20. A seventeen-member committee must elect one
of four candidates: R, S, T, or W. See the preference
schedule below.
First choice
Second choice
Third choice
Fourth choice
Number of Members
6
4
3
4
R
S
T
W
S
R
S
T
T
T
R
S
W
W
W
R
Is it possible to manipulate the results of a
sequential pairwise election?
No. S will always win.
A seventeen-member committee must elect one of
four candidates: R, S, T, or W. See the preference
schedule below.
First choice
Second choice
Third choice
Fourth choice
Number of Members
6
4
3
4
R
S
T
W
S
R
S
T
T
T
R
S
W
W
W
R
In a plurality runoff election, candidate S wins.
What would happen if the four voters who prefer
W insincerely voted for T instead? Is this in their
best interests?
T and R would face off in the runoff, and R would
win. In this case, their least favorite candidate
would win instead of a higher ranked alternative.
A seventeen-member committee must elect one of
four candidates: R, S, T, or W. See the preference
schedule below.
First choice
Second choice
Third choice
Fourth choice
Number of Members
6
4
3
4
R
S
T
W
S
R
S
T
T
T
R
S
W
W
W
R
In a plurality runoff election, candidate S wins.
What would happen if the four voters who prefer
T insincerely voted for S instead? Is this in their
best interests?
S and R would face off in the runoff, but S would
again win. They cannot force a win for their first
choice, but they can show allegiance to their
second choice and eventual winner.
An 11-member committee must choose one of the
four applicants, K, L, M, and N, for membership on
the committee.
Number of Members
6
2
3
First choice
K
M
M
Second choice
L
L
N
Third choice
N
K
L
Fourth choice
M
N
K
The committee members have preferences among the
applicants as given in the table. If the committee uses
pairwise sequential voting with the agenda K, L, M, N,
applicant K wins. Can the three voters who least prefer K
vote strategically in some way to change the outcome to
one they find more favorable? Why or why not?
No. The six voters who most prefer applicant K
represent a majority of the committee. No matter
how the three voters rank the applicants, K will
win.
24. An 11-member committee must choose one of the
four applicants, K, L, M, and N, for membership on the
committee.
Number of Members
6
2
3
First choice
K
M
M
Second choice
L
L
N
Third choice
N
K
L
Fourth choice
M
N
K
The committee members have preferences among
the applicants as given in the table. If the committee
uses pairwise sequential voting with the agenda K,
L, M, N, applicant K wins. Is it possible that another
agenda will yield a different winner?
No. The six voters who most prefer applicant K
represent a majority of the committee. No matter
how the voters are ordered, K will win.
25. An 11-member committee must choose one of the four
applicants, K, L, M, and N, for membership on the
committee.
Number of Members
6
2
3
First choice
K
M
M
Second choice
L
L
N
Third choice
N
K
L
Fourth choice
M
N
K
The committee uses the Borda count method. The
committee members have preferences among the
applicants as given in the table. Who wins the
election? Can the group of three voters favorably
impact the results through insincere voting?
K currently wins. Yes. For example, by
exchanging L and N, L will win instead.
An 11-member committee must choose one of the
four applicants, K, L, M, and N, for membership on
the committee.
Number of Members
6
2
3
First choice
K
M
M
Second choice
L
L
N
Third choice
N
K
L
Fourth choice
M
N
K
The committee uses the Borda count method. The
committee members have preferences among the
applicants as given in the table. Who wins the
election? Can the group of two voters favorably
impact the results through insincere voting?
K currently wins. Yes. For example, by
exchanging L and M, L will win instead.
An 11-member committee must choose one of the
four applicants, K, L, M, and N, for membership on
the committee.
Number of Members
6
2
3
First choice
K
M
M
Second choice
L
L
N
Third choice
N
K
L
Fourth choice
M
N
K
The committee uses the Borda count method. The committee
members have preferences among the applicants as given in
the table. Suppose the group of six suspect that the group of
two intends to insincerely exchange M and L in their rankings.
Can the group of six counteract in order to protect K as the
winner?
Yes. If the group of two exchange M and L and
the group of six exchange L and N, K will still win.
There are 18 delegates to a political party's convention at
which four people, A, B, C, and D, have been nominated
as the party's candidate for governor. The delegates'
preference schedule is shown below. If the party uses a
Borda count, candidate B would be elected. Can the four
voters who most prefer C vote strategically in some way
to change this outcome to one they would find more
favorable? Why or why not?
First choice
Second choice
Third choice
Fourth choice
Number of Delegates
8
9
4
A
B
C
B
A
B
C
D
A
D
C
D
No. They do not have enough votes to make C win
under any possible ranking. Their second choice is B,
who is the winner.
Consider an 11-member committee that must
choose one of three alternatives, X, Y, or Z, using
the Hare system. Their schedule of preferences is
shown below.
Number of Voters
5
4
2
First choice
Z
X
Y
Second choice
Y
Y
X
Third choice
X
Z
Z
Who wins? Is it possible for the group of five
voters to change the outcome in a way that
would benefit them?
X wins. If the group of five voters exchange their
rankings of Y and Z, then Y wins.
30. Consider an 11-member committee that must
choose one of three alternatives, X, Y, or Z, using the
Hare system. Their schedule of preferences is shown
below.
Number of Voters
5
4
2
First choice
Z
X
Y
Second choice
Y
Y
X
Third choice
X
Z
Z
Who wins? Is it possible for the group of two
voters to change the outcome in a way that
would benefit them?
X wins. The group of two voters cannot change
the outcomes by insincerely changing their
preference ordering.
Consider an 11-member committee that must
choose one of three alternatives, X, Y, or Z, using
the Hare system. Their schedule of preferences is
shown below.
Number of Voters
5
4
2
First choice
Z
X
Y
Second choice
Y
Y
X
Third choice
X
Z
Z
The committee suspects that the group of five
plans to insincerely reorder their preferences as
Y, Z, X. How can the group of four respond?
If the group of five exchange Y and Z, then Y
wins. The group of four cannot retaliate against
this change.
Twenty-nine voters must choose from among three
alternatives, A, B, and C, using the Borda count
method. The voters' preference schedules are
shown below.
First choice
Second choice
Third choice
Number of Voters
12
8
6
B
C
A
C
A
B
A
B
C
3
C
B
A
Who wins Borda count? Can the group of six
voters change their preference list to produce an
outcome they like better?
C wins. If the group of six exchange A and B in
their rankings, then B will win instead.
Twenty-nine voters must choose from among three
alternatives, A, B, and C, using the Borda count
method. The voters' preference schedules are
shown below.
First choice
Second choice
Third choice
Number of Voters
12
8
6
B
C
A
C
A
B
A
B
C
3
C
B
A
Suppose the group of voters anticipate that the
group of six plan to insincerely rank B above A.
How can the remaining voters respond in their
own rankings?
B is the winner. The remaining voters are unable
to alter their rankings in order to elect a more
favored candidate.
Given the weighted voting system [30:
20, 17, 10, 5], list all winning
coalitions.
• (A, D) (A, C) (B, C, D) (A, B, C)
(A, B, D) (A, C, D) (A, B, C, D)
35. Given the weighted voting system
[51: 45, 43, 7, 5], list all blocking
coalitions.
• (A, B) (A, C) (A, B, C) (A, B, D)
(A, C, D) (A, D) (B, C) (B, C, D)
(A, B, C, D)
Given the weighted voting system [30:
20, 17, 10, 5], list all blocking
coalitions.
• (A, B) (A, C) (A, B, C) (A, B, D)
(A, C, D) (B, C, D) (A, B, C, D)
(A, D) (B, C)
Given the weighted voting system [5:
3, 2, 1, 1, 1], find which voters of the
coalition { A, C, D, E} are critical.
Since the coalition { A, C, D, E} has
one extra vote, the only member
who is critical is voter A with weight
3.
Given the weighted voting system [7:
4, 1, 1, 1, 1, 1], find the Banzhaf power
index for each voter.
(32, 20, 20, 20, 20, 20)
Calculate the Shapley-Shubik power
index for the weighted voting system
[30: 20, 17, 10, 5].
5
( ,
12
1
,
4
1
,
4
1
)
12
40. Given the weighted voting system
[16: 3, 9, 4, 5, 10], calculate the
Banzhaf power index for each voter.
(4, 8, 4, 4, 8)
A weighted voting system has 12
members. How many distinct
coalitions are there in which exactly
six members vote yes?
120