practice_midterm1_solutions
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University of Maryland *
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Course
130
Subject
Mathematics
Date
Apr 3, 2024
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6
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Math 130 Practice Midterm 1 Spring 2024
You have 50 minutes to complete the test.
This exam contains 3 two-sided pages and 4 questions. If you have any questions, raise your hand and wait for the
proctor to come to your seat.
Write your final solutions on the test papers. Solutions presented with no supporting work may receive no credit.
If you need extra space, use the extra page at the back. Clearly indicate on the question when you use paper at the
back, and clearly indicate at the back which question you are answering. You may use the the backs of the pages
and
/
or the last page of the exam to write your scratch-work. Please ask the proctor if you need more paper.
Recall that you are allowed to use the results from the book and the class. When doing so, please be sure to write
down the statements of the Theorems you are using.
YOU MAY NOT USE NOTES, CELL PHONES, OR LAPTOPS AT ANY TIME DURING THE TEST
PERIOD. When going to the bathroom, please leave your phone on the table.
You’ve got this!
PLEASE PRINT YOUR NAME HERE IN BLOCK CAPITALS:
PLEASE WRITE YOUR BNUMBER HERE:
Page 1
Math 130
Quiz
1. (25 points) The Binghamton anime club is holding an election to determine it’s o
ffi
cers. There are 5 candidates
labelled A, B, C, D, E. The highest ranked candidate will be president, the 2nd highest will be vice-president,
and the third ranked will be treasurer. The ballots are as follows:
Ballot
1.
D
2.
E
3.
C
4.
A
5.
B
Ballot
1.
C
2.
B
3.
E
4.
D
5.
A
Ballot
1.
B
2.
C
3.
A
4.
D
5.
E
Ballot
1.
A
2.
D
3.
B
4.
C
5.
E
Ballot
1.
B
2.
E
3.
A
4.
C
5.
D
Ballot
1.
D
2.
E
3.
C
4.
A
5.
B
Ballot
1.
D
2.
E
3.
C
4.
A
5.
B
Ballot
1.
D
2.
E
3.
C
4.
A
5.
B
Ballot
1.
B
2.
C
3.
A
4.
D
5.
E
Ballot
1.
B
2.
E
3.
A
4.
C
5.
D
Ballot
1.
C
2.
B
3.
E
4.
D
5.
A
Ballot
1.
C
2.
B
3.
E
4.
D
5.
A
Ballot
1.
C
2.
B
3.
E
4.
D
5.
A
Ballot
1.
B
2.
C
3.
A
4.
D
5.
E
Ballot
1.
D
2.
E
3.
C
4.
A
5.
B
(a) (10 points) Compile the preference schedule for the above preference ballots.
Number of ballots
Choice
5
4
3
1
2
1
D
C
B
A
B
2
E
B
C
D
E
3
C
E
A
B
A
4
A
D
D
C
C
5
B
A
E
E
D
(b) (15 points) Use the
plurality by elimination method
to rank all the candidates and determine who is
president, vice-president and treasurer.
In the first round E has the least first place votes (0 votes) so they are removed from the election.
The first place votes now stand at A-1 B-5 C-4 D-5.
In the second round A has the least first place votes (1 vote) so they are removed from the election.
The first place votes now stand at B-5 C-4 D-6.
In the third round C has the least first place votes so they are removed from the election.
The first place votes now stand at B-9 D-6.
Now D is removed and B is first.
Thus the ranking in descending order is B, D, C, A, E.
B is president, D is vice-president, C is treasurer.
Page 2
2. (25 points) Consider the following preference schedule for an election, where ”A” refers to ”Ayanami Rei”,
”B” to ”Bishamonten”, ”C” to ”Chinatsu”, and ”D” to ”DIO”.
Number of ballots
Choice
2
4
4
3
4
1
C
D
D
C
C
2
A
C
B
A
D
3
D
B
A
D
B
4
B
A
C
B
A
(a) (10 points) Is there a candidate with a majority in the above election? Justify your answer.
Yes, Chinatsu has 9 of the available 17 first place votes. So Chinatsu has a majority.
(b) (8 points) Which of the following four voting methods satisfy the majority criterion in
all
possible elec-
tions with any number of candidates and voters:
plurality
,
basic borda count
,
plurality with elimina-
tion
,
pairwise comparisons
.
plurality
,
plurality with elimination
,
pairwise comparisons
.
(c) (17 points) Determine the winner of the above election using all four of the voting methods mentioned
in the previous question. Justify your answer. [Hint: you can save some work using your answers to the
previous questions]
From our previous answers, since Chinatsu has a majority, we know that Chinatsu wins the election by
each of the
plurality
,
plurality with elimination
, and
pairwise comparisons
methods. It remains to
check basic borda count.
1. A has 3(2
+
3)
+
2(4)
+
1(4
+
4)
=
31 points
2. B has 3(4)
+
2(4
+
4)
+
1(2
+
3)
=
33 points
3. C has 4(2
+
3
+
4)
+
3(4)
+
1(4)
=
52 points
4. D has 4(4
+
4)
+
3(4)
+
2(2
+
3)
=
54 points
So DIO wins by basic borda count.
Page 3
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3. (20 points) Consider the weighted voting system [9 : 6
,
5
,
4
,
3] with voters
A
,
B
,
C
,
D
in the given order.
(a) (15 points) List all the coalitions and their weights. Determine which are winning coalitions and which
voters are critical voters in each such coalition.
•
{
D
}
has weight 3, this is not a winning coalition (hence no critical voters).
•
{
C
}
has weight 4, this is not a winning coalition (hence no critical voters).
•
{
C, D
}
has weight 7, this is not a winning coalition (hence no critical voters).
•
{
B
}
has weight 5, this is not a winning coalition (hence no critical voters).
•
{
B, D
}
has weight 8, this is not a winning coalition (hence no critical voters).
•
{
B, C
}
has weight 9, this is a winning coalition, B and C are both critical voters.
•
{
B, C, D
}
has weight 12, this is a winning coalition, B and C are the only critical voters.
•
{
A
}
has weight 6, this is not a winning coalition (hence no critical voters).
•
{
A, D
}
has weight 9, this is a winning coalition, both A and D are critical voters.
•
{
A, C
}
has weight 10, this is a winning coalition, both A and C are critical voters.
•
{
A, C, D
}
has weight 13, this is a winning coalition, only A is a critical voter.
•
{
A, B
}
has weight 11, this is a winning coalition, both A and B are critical voters.
•
{
A, B, D
}
has weight 14, this is a winning coalition, only A is a critical voter.
•
{
A, B, C
}
has weight 15, this is a winning coalition, there are no critical voters.
•
{
A, B, C, D
}
has weight 18, this is a winning coalition, there are no critical voters.
(b) (5 points) Find the power index of all voters (leave your answers as fractions).
The number of critical voter occurrences is: 1
+
2
+
1
+
2
+
2
+
2
+
2
=
12.
The power indexes are
1. A: 5
/
12,
2. B:3
/
12,
3. C:3
/
12,
4. D:1
/
12.
Page 4
4. (20 points) Consider the weighted voting system [
q
: 7
,
4
,
2
,
1] with voters
A
,
B
,
C
,
D
in the given order.
(a) (8 points) Which of the following values of
q
satisfy the quota restriction?
q
=
4
,
q
=
7
,
q
=
10
,
q
=
14
,
q
=
15. Justify your answer.
The quota restriction is satisfied for integers
q
satisfying 7
=
7
+
4
+
2
+
1
2
<
q
≤
7
+
4
+
2
+
1
=
14. So among
the given options,
q
=
10 and
q
=
14 are the only options which satisfy the quota restriction.
(b) (7 points) For each of the values of
q
satisfying the quota restriction found in question a), determine if
the system has voter(s) with veto power? List the veto power voters in each case.
This would mean that a voter has the power to block the quota being satisfied alone.
•
If
q
is 10, then only voter A can do this.
•
If
q
is 14, then every voter has veto power.
(c) (5 points) Suppose that we have a Law firm with five partners
A
,
B
,
C
,
D
,
E
which accepts proposed
changes only when either A and B approve or at least 3 partners approve. Express an equivalent system
in terms of weights and quotas.
This is perhaps the most simple answer: [6 : 3
,
3
,
2
,
2
,
2].
There are infinitely many other acceptable answers. For example [23 : 12
,
11
,
10
,
9
,
8].
Page 5
Extra paper page.
Page 6
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