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Test 1 Review
MATH 146
1. Four students are running for president of the Math club: Antoine (A), Betty (B), Camille (C)
and Don (D). The voters were asked to rank all candidates. The resulting preference table for
this election is given below.
Number of Votes
1st
2nd
3rd
4th
a)
b)
c)
d)
e)
f)
g)
15
B
A
C
D
11
C
A
D
B
9
D
C
A
B
5
A
D
C
B
1
C
D
A
B
How many students voted in the election?
How many students selected for each candidate?
Who is selected president using the plurality method?
Who is selected president using Borda Count Method?
Find the winner of the election for president of the club using plurality-with-elimination.
Who is the president of the club using the Pairwise Comparisons Method?
Does it have the same winner with different methods from above? What conclusion we
can make with these voting method?
2. The following ballots were used to determine which food to serve at a class party. The
choices were between Chef Salad, Burritos, Hamburgers and Pizza. Students voted by
ranking their choices in order of preference, with '1' being the favorite and '4' being the least
popular.
Ballot
Ballot
Ballot
Ballot
Ballot
Ballot
Ballot
Ballot
Ballot
1.CS
2 Burr
3 Hamb
4 Pizza
1 Hamb
2.Pizza
3 Burr
4 CS
1. CS
2. Burr
3 Hamb
4 Pizza
1.CS
2.Burr
3 Hamb
4 Pizza
1.Pizza
2.Burr
3.Hamb
4 CS
1 Burr
2 Hamb
3.Pizza
4 CS
1.CS
2 Burr
3 Hamb
4 Pizza
1. Pizza
2. Burr
3. Hamb
4 CS
1 Hamb
2.Pizza
3.Burr
4 CS
Ballot
Ballot
Ballot
Ballot
Ballot
Ballot
1 Burr
2 Hamb
3. Pizza
4 CS
1.CS
2 Burr
3 Hamb
4 Pizza
1 Hamb
2.Pizza
3 Burr
4 CS
1.Pizza
2.Burr
3.Hamb
4 CS
1.CS
2 Burr
3 Hamb
4 Pizza
a)
b)
c)
d)
Make preference schedules to represent the voting results.
Is there a majority winner? Explain.
Which food is the plurality winner? How did you determine that answer?
Suppose that the election is to decided under the Borda count method,
find the winner.
e) Suppose that a plurality-with-elimination is used to determine the winner,
which food would finally win the election? How did you determine that answer?
f) Which food would be the winner if the method of Pairwise Comparisons is used?
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3. MATH 146 class voted on where to take a trip? To a beach, a
trip to the mountains, a ski trip, or a Disney world trip. The
preference schedule on the right resulted from the ballots.
a) Determine the winner using the Borda Count Method.
b) Is there a winner from the majority method?
c) Borda Count method violate what criteria?
1st
choice
2nd choice
3rd choice
4th choice
10
B
M
S
D
5
S
M
D
B
4
D
S
M
B
4. Find the winner of the election given by the following preference schedule under the method
of pairwise comparisons.
Number of voters
1st Choice
2ND Choice
3rd Choice
4th Choice
5th Choice
5
A
B
C
D
E
3
A
D
B
C
E
5
C
E
D
A
B
3
D
C
B
E
A
2
D
C
B
A
E
3
B
E
A
C
D
5. Find the winner of the election given by the following preference schedule under the method
of pairwise comparisons.
Number of voters
1st Choice
2nd Choice
3rd Choice
4th Choice
3
A
B
C
D
4
A
B
D
C
9
A
C
B
D
9
B
C
D
A
2
B
A
C
D
5
B
C
A
D
8
C
D
B
A
3 12
C D
A C
D A
B B
6. Find the winner of the election given by the following preference schedule under the method
of pairwise comparisons.
Number of voters
1st Choice
2nd Choice
3rd Choice
4th Choice
8
B
A
C
D
6
A
D
B
C
5
C
D
B
A
5
D
A
C
B
2
D
A
B
C
7. An election is held to choose the president of the class. The
candidates are A (Andrew), M (Megan), K (Keaten), and R (Robby).
The preference schedule is listed on the right.
a) Which candidate would be the winner if using the plurality-withelimination method?
b) Two of the voters at the last column decided to change their votes
by moving Megan to their first choice. So, these two votes now
change their ballot
1st
5
6
8
2
A
K M A
2nd K M
R
3rd M A
A M
4th
K
R
R
R
K
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2
A
from
R
2
M
to
A
M
R
K
K
Which candidate would be the winner if using the plurality-with-elimination method?
c) From a) and b) above, imply a violation of which fairness criterion? Why? Explain
clearly.
8. A local book club with five active members is voting to decide which book to read next.
Three book choices have been nominated: a mystery (M), a historical novel (H), and a
science fiction fantasy(S). The members’ preferences for the three choices are shown at the
right.
a) Which of the three books would be selected using the plurality with
2 1
elimination method?
st
1
M H
b) Suppose that one member points out that the science fiction book is
2nd S M
not yet available in paperback, so the club decides to eliminate this
3rd H S
book from consideration. Would this affect the outcome of the
selection process? Which is the winner?
c) The results from a) and b) imply a violation of which fairness criterion? Why? Explain
clearly.
2
S
H
M
9. Consider a 4-person panel that makes decisions by using weighted voting. The four people
have 8, 4, 3 and 2 votes respectively.
a)
b)
c)
d)
Explain why it would not make sense to set the quota at 18 votes.
Give an example of a quota which would give one person veto power.
Give an example of a quota which will make one person be a dummy.
Use Banzhaf Power Index to find the power distribution for this voting system
[13: 8, 4, 3, 2]
10. Consider the weighted voting system [22:10, 10, 10, 10, 1],
a) Are there any dummies? Explain your answer.
b) Find the Banzhaf power distributions of this weighted voting system.
11. a) How many possible coalitions are there if we use Banzhaf Power Index for n players?
b) How many possible sequential coalition are there if we use Shapley–Shubik Power Index
for n players?
12. Find the Shapley-Shubik Power Distribution of [8: 7, 6, 2] weighted voting.
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13. Find the Shapley-Shubik Power Distribution of the weighted voting system
[60: 32, 31, 28, 21]
14. In the weighted voting system [31: 12, 8, 6, 5, 5, 5, 2], find
a) the total number of players.
b) the total number of votes.
c) the weight of p5 .
d) any player(s) who has a veto power.
e) any player(s) who is a dummy.
15. For the weighted voting system [48: 32, 16, 8, 4, 2, 1], determine which players if any,
a) are dictators.
b) have veto power.
c) are dummies?
16. Four players (Abe, Betty, Cory and Dana) must divide a cake among themselves. Suppose
the cake is divided into 4 slices (s1, s2, s3 and s4). The values of the entire cake and of each of
the 4 slices in the eyes of each of the players are shown in the following table:
Player Whole Cake
s1
s2
s3
s4
Abe
$15.00
$3.00
$5.00
$5.00
$2.00
Betty
$18.00
$4.50
$4.50
$4.50
$4.50
Cory
$12.00
$4.00
$3.50
$1.50
$3.00
Dana
$10.00
$2.75
$2.40
$2.45
$2.40
a) Using the four given slices (and the worth that each player assigns them), find a fair
division for the cake.
b) Explain why the division is considered “fair”.
17. Six players want to divide a cake fairly using the lone-divider method. The divider cuts the
cake into 6 slices { s1 , s2 , s3 , s4 , s6 }, and the choosers make the following declarations:
Chooser 1: { s2 , s3 , s5 }
Chooser 2: { s1 , s5 , s6 }
Chooser 3: { s3 , s5 , s6 }
Chooser 4: { s2 , s3 }
Chooser 5: { s3 }
a) Describe a fair division of the cake.
b) Explain why the answer in a) above is the only possible fair division of the cake.
18. Three players {Alex, Betty, and Cindy} must divide a cake among themselves. Suppose the
cake is divided into three slices { s1 , s2 , s3 }. The following table shows the percentage of the
value of the entire cake that each slice represents to each player.
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s1
s2
s3
Alex
30%
40%
30%
Betty
35%
25%
40%
Cindy 33⅓ % 33⅓ % 33⅓ %
a)
b)
b)
c)
Who is the divider? Why? Explain clearly.
Indicate all possible slices are fair shares to Alex, Betty and Cindy.
Describe a fair division of the cake for each person.
Explain why the division is considered “fair”.
19. Three players (A, B and C) wish to divide up 5 items using the method of sealed bids. Their
bids on each of the items are given in the following table:
Item
1998 Ford Explorer
Share in a Hawaii condominium
Harley-Davidson motorcycle
Fur coat (mink stroller)
Diamond necklace and earrings
A
$ 15,000
$ 24,000
$ 17,000
$ 16,000
$ 18,000
B
$ 11,000
$ 15,000
$ 18,000
$ 16,000
$ 24,000
C
$ 22,000
$ 33,000
$ 15,000
$ 18,000
$ 20,000
Describe the final outcome of this fair-division problem.
20. Two players, Evan and Justin, wish to dissolve their business partnership using the method of
sealed bids. Evan bids $15,000 and Justin bids $23,000.
a) How much are Evan and Justin’s fair shares worth?
b) How much is the surplus after the allocations are made?
c) What is the final decision for each of them?
21. Suppose a state has 5 counties with populations of 8345, 581, 463, 391 and 220 people. It is
decided that a total of 20 people should be seated in the state legislature.
a) What is the standard divisor? What does it mean? Write the conclusion in sentence with
units.
b) How should the delegates be apportioned? Fill in the following table with the
apportionment that would be produced by the Hamilton, Jefferson, Adams and Webster
methods.
c) Based on the different methods above, which method violate Quota Rule? What county
is affected?
d) Time passes, and each county gains an additional 200 people. Now, the total population
is 11,000, the apportionment is still 20 seats. The new populations are shown in the table
below. Find the apportionment using the Hamilton’s method for the new population
totals.
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Hamilton’s
County Population Method
A
8,545
B
781
C
663
D
591
E
420
e) Compare the new apportionment with the one that used the Hamilton’s Method when the
population was 10,000 people. What strikes you as being odd about this new result?
What is the name of Paradox that it has violated?
22. The Central City school district receives a grant to purchase 50 new laser printers to be
distributed among the five schools in the district. The following table shows each school’s
population.
School Population Standard Quota Lower Quota Hamilton's Method
A
210
B
165
C
160
D
175
E
190
Total
a) What is the standard divisor. Translate to complete sentence with units.
b) Find the apportionment for each country using Hamilton's Method.
23. A small country with a population of 10,000,000 consists of four states. There are 250 seats
in the legislature that need to be apportioned among the four states. The population of each
state is shown below.
State Population Standard Quota
A
3,250,000
B
1,750,000
C
2,265,000
D
2,735,000
Total
a)
b)
c)
d)
What is the standard divisor? Write the conclusion in a complete sentence.
Find each state’s apportionment using Hamilton’s Method.
Find each state’s apportionment using Jefferson’s Method.
Find each state’s apportionment using Adams’ Method.
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e) Find each state’s apportionment using Webster’s Method.
24. Five friends had shared ownership of a used car business since they finished business school.
After 5 years, they decide to start his or her own used car business and do not wish to
continue their partnership. They decide to use the method of markers to divide up the cars
and other equipment that they have purchased jointly and have arranged an array of 20 items
as shown below.
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
C2 A2 D2 B2 C3 A3 B3 C4 D4 A4
E2
E3
B4
D3
E4
a) Describe the allocation of items to each partner. Who get which items?
b) Which items are left over?
C1 A1 B1
D1 E1
25. Four players are dividing the array of 15 items shown in the following using the method of
markers. The players’ bids are indicated below.
1
2
3
4
5
6
8
A2
9
10
11
12
13
14
15
D2 A3 C3
B3
D3
a) Describe the allocation of items to each partner. Who get which items?
b) Which items are left over?
C1
B1 C2 A1
7
B2 D1
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MATH 146
Test 1 Review Answer Key
1. a) 41 students
b) B =15, C = 12, A = 5, D = 9
c) Betty is the winner.
d) B = 86, A = 118, C = 115, D = 91 Antoine is the winner.
e) D = 26, B = 15 Don is the winner.
f) A = 2, B = 0, C = 3, D = 1 Camille is the winner.
g) no, each method has different winner. There is no method will satisfy the fairness criteria.
2.
a)
No. of voters
6
3
3
2
1st choice
C
H
P
B
2nd choice
B
P
B
H
3rd choice
H
B
H
P
th
4 choice
P
C
C
C
b) No, we need at least 8 votes. Explain.
c) C = 6, H = 3, P = 3, B = 2 Chef Salad is the winner.
d) C= 32, B = 41, H = 36, P = 31 Burrito is the winner.
e) H = 8 Hamburger is the winner.
f) B = 3, H = 2, P = 1, C = 0, Burrito is the winner.
3. a)
b)
c)
B = 49, M = 53, S = 52, D = 36. So, Mountain trip is the winner.
B has the majority vote so beach trip is the winner.
Borda Count violate the majority criterion.
4. Candidate A is the winner. A = 3 points, B = 2 points, C = 2 points, D = 2 points, E = 1 point.
5. Candidate A is the winner. A = 1 point, B = 0 point, C = 3 points, D = 2 points
6. Candidate A is the winner. A = 2.5 points, B = 1.5 points, C = 0.5 point, D = 1.5 points.
7. a) Megan is the winner
b) Keaten is the winner
c) Monotonicity Criterion. If Megan is the winner of an election and in a reelection, the
only changes in the ballots are changes that Megan to be the first rank, then Megan
should remain a winner of the election. However, from b) above, Megan is not a winner
any more.
8. a) mystery
b) yes, history
c) violate Independence-of Alternatives criterion (IIA). If mystery is the winner of an
election and in a recount one of the nonwinning candidates (science fiction) withdraws,
then, mystery should still be a winner of the election.
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9. a)
b)
c)
d)
the highest possible vote is 17 votes.
[10: 8, 4, 3, 2] will make P1 a veto power.
[11: 8, 4, 3, 2] will make P4 a dummy.
[13: 8, 4, 3, 2] p1  40%, p2  20%, p3  20%, p4  20%
10. a) p5 is the dummy. Explain.
b) p1  25%, p2  25%, p3  25%, p4  25%, P5  0% . As you can see, P5 weighs
nothing. So, P5 is a dummy.
11. a) 2n  1 where n = total number of players.
b) n !
where n = total number of players.
12.
P1 
2
 33.3%
6
2
P2   33.3%
6
13.
P1 
10
 41.7%
24
P2 
6
 25%
24
P3 
P3 
2
 33.3%
6
6
 25%
24
P4 
2
 8.3%
24
14. a) Total 7 players.
b) Total 43 votes.
c) The weight for p5 is 5 votes.
d) no player has a veto power.
e) no dummy.
15. a) There is no dictator.
b) P1 , P2 have veto power.
c) P3 , P4 , P5 , P6 are dummies.
16. Abe { s2 , s3 }, Betty is the divider { s1 , s2 , s3 , s4 }, Cory { s1 , s2 , s4 } and Dana { s1 }.
a) Dana { s1 } = $2.75, Cory { s2 }= $3.50, Abe { s3 } = $5.00, Betty { s4 }= $4.50
b) Everyone gets more than “fair” share. Abe get more than $3.75, Betty gets $ 4.50, Cory
gets more than $3 and Dana gets more than $2.50.
17. a) Chooser 5 { s3 } Chooser 4 { s2 } Chooser 1{ s5 }, Chooser 3 { s6 }, Chooser 2 { s1 }
Divider { s4 }
b) Explain
18. a) Cindy is the divider since she divided each slice equally.
b) Alex { s2 }, Betty { s1 , s3 }, Cindy { s1 , s2 , s3 }
c) Alex { s2 } = 40%, Cindy { s1 } = 33⅓ %, Betty { s3 } = 40%
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d) Every person has more than a fair division which is equal or more than 33⅓ %, Alex has
40%, Cindy has 33⅓ %, and Betty has 40% of the cake.
19.
Item
1998 Ford Explorer
Share in a Hawaii condominium
Harley-Davidson motorcycle
Fur coat (mink stroller)
Diamond necklace and earrings
Total bids
Fair Shares
Allocation
A
$ 15,000
$ 24,000
$ 17,000
$ 16,000
$ 18,000
$90,000
$30,000
B
C
$ 11,000
$ 22,000
$ 15,000
$ 33,000
$ 18,000
$ 15,000
$ 16,000
$ 18,000
$ 24,000
$ 20,000
$84,000
$108,000
$28,000
$36,000
Motorcycle,
Ford, Condo,
Diamond
Fur Coat
Pay/Received
Rev $30,000
Pay $14,000
Pay $37,000
Surplus
Rev $7,000
Rev $7,000
Rev $7,000
Final
Rev $37,000
Pay $7,000
Pay $30,000
A = {received $37,000} B = { got motorcycle, diamond and paid $7000}
C = {got Ford, Condo, Fur Coat and Paid $30,000}
20.
Item
Even
Justin
Business
$ 15,000
$ 23,000
Total bids
$15,000
$23,000
Fair Shares
$7,500
$11,500
Allocation
Business
Pay/Received
Rev $7,500
Pay $11,500
Surplus
Rev $2,000
Rev $2,000
Final
Rev $9,500
Pay $9,500
a) For Evan, the fair share is to $7,500. For Justin, the fair share is $11,500.
b) For each of them, they would receive $2,000 for the surplus.
c) For Evan {received $9,500}, For Justin {got business and paid $9,500}
21. a) standard divisor = 500. For 1 seat in the state legislature, it will serve 500 people.
b)
Hamilton
Hamilton’s
¸ 500 Lower Upper
County Population
Quota Quota
Method
A
8345
17.29
17
18
17
B
581
1.162
1
2
1
C
463
0.926 +1 0
1
1
D
391
0.782+1 0
1
1
E
220
0.44
1
0
0
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Hamilton’s Method: A = 17, B = 1, C = 1, D = 1, E = 0
Jefferson Method: Modified divisor, Lower, Use
Lower Quota
(Use modified divisor: 440, 450 or 460)
¸ 450 Lower Jefferson’s
County Population
Quota
Method
A
8345
18.544
18
18
B
581
1.291
1
1
C
463
1.028
1
1
D
391
0.869
0
0
E
220
0.489
0
0
Jefferson’s Method: A= 18, B = 1, C = 1, D = 0, E = 0
Adam's Method: Modifier divisor, higher. Use
upper quota
(Use modified divisor: = 560 or 570 or 580)
¸ 570 Upper Adam’s
County Population
Quota
Method
A
8345
14.640
15
15
B
581
1.019
2
2
C
463
0.724
1
1
D
391
0.685
1
1
E
220
0.350
1
1
Adams’ Method: A = 15, B = 2, C = 1, D = 1, E = 1
Webster's Method: Modifier divisor: random, Use
normal rounding rule
(Use modified divisor: 480 or 490 or 500)
Normal
Webster’s
¸ 500
County Population
Rounding
Method
A
8345
16.69
17
17
B
581
1.162
1
1
C
463
0.926
1
1
D
391
0.782
1
1
E
220
0.44
0
0
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Webster’ Method: A = 17, B = 1, C = 1, D = 1, E = 0
c) Adam’s Method: State A violate Lower Quota rule. State A should be between
17 and 18 but it was 15 using Adam's Method. Also State E is upset because it has 0 seat
with Hamilton's Method, Jefferson's Method, and Webster's Method. State D has 0 seat
under Jefferson's Method.
d) Standard Divisor = 550. For every 550 population, there is only one seat in the state legislature.
Hamilton’s
¸ 550 Lower Upper
County Population
Quota Quota
Method
A
8545 15.535 15
18
16
B
781
1.42
1
2
1
C
663
1.205
1
1
1
D
591
1.074
1
1
1
E
520
0.763
0
1
1
Hamilton’s Method: A = 16, B = 1, C = 1, D =1, E =1.
e) After increased 200 people in County A loses 1 seat. For County E, it gains 1 seat after
the population increased. County D did not gain any seat even the population increased.
County C did not gain any seat after the population increased. County B did not gain any
seat after the population gains. This is the population paradox.
22. standard division = 18. For every 18 people, there is one laser printer.
School Population Standard Quota Lower Quota Hamilton's Method
¸ 18
A
210
11.666
11
12
B
165
9.167
9
9
C
160
8.889
8
9
D
175
9.722
9
10
E
190
10.556
10
10
Total
47+3
50
Use Hamilton's method: A = 12, B = 9, C = 9, D = 10, E =10
23. a) standard divisor: 10,000,000/250 = 40,000. For every 40,000 population, there is one
seat in the legislature.
b)
Hamilton's Method: Standard divisor: 40,000
State Population Standard Quota Lower Upper
Hamilton's
Quota Quota
Method
A
3,250,000
81.25
81
82
81
B
1,750,000
43.75
43
44
44
12
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C
2,265,000
56.625
D
2,735,000
68.375
Total 10,000,000
Hamilton’s A = 81, B = 44, C = 57, D = 68
56
68
248
57
69
57
68
250
Jefferson's Method: Use Modified divisor 39,700
State Population Standard Quota Lower
Jefferson's
Quota
Method
A
3,250,000
81.863
81
81
B
1,750,000
44.081
44
44
C
2,265,000
57.053
57
57
D
2,735,000
68.892
68
68
Total 10,000,000
250
250
Jefferson’s Method: A = 81, B = 44, C = 57, D = 68
Adam's Method: Use Modified divisor 40,300
State Population Standard Quota Upper
Quota
A
3,250,000
80.645
81
B
1,750,000
43.424
44
C
2,265,000
56.203
57
D
2,735,000
67.866
68
Total 10,000,000
250
Adam's
Method
81
44
57
68
250
Adams’ Method: A = 81, B = 44, C = 57, D = 68
Webster's Method: Use Modified divisor 40,000
State Population Standard Quota Normal
Rounding
A
3,250,000
81.25
81
B
1,750,000
43.75
44
C
2,265,000
56.625
57
D
2,735,000
68.375
68
Total 10,000,000
250
Adam's
Method
81
44
57
68
250
Webster’s Method: A = 81, B = 44, C = 57, D = 68
24a) A = {10, 11, 12, 13}, B = {18, 19, 20}, C = {1, 2, 3}, D = {15, 16}, E = {6, 7, 8}
b) Left over= {4, 5, 9, 14, 17}
25a)
b)
C = {1, 2, 3}, A = {7}, D = {11}, B = {15}
The left over are {4, 5, 6, 8, 9, 10, 12, 13, 14}
13