Deck 13: Fair Division

Question 1

Suppose that Bob and Carol want to take turns, using the bottom-up strategy, to allocate several mathematics and statistics textbooks currently held jointly. Their ranked preferences are listed below: If Carol chooses first, what is the division of the objects?

Answer 1

Carol selects trigonometry, algebra, and statistics.

Question 2

The administration and the student body at a local college are at odds over several issues. The relative importance of each issue to each group is as follows: Use the adjusted winner procedure to determine a fair resolution.

Answer 2

The administration gets its way on the mandatory meal plan, and 11/12 of its way with regard to weekday fraternity parties. The student body gets its way on the rest of the issues.

Question 3

Henry and Lisa inherit a house. If their monetary bids on the house are $135,000 and $114,000, respectively, what is the fair division arrived at by the Knaster inheritance procedure?

Answer 3

Henry receives the house and gives Lisa $62,250.

Question 4

After having been roommates for four years at college, Alex and Bob are moving on. Several items they have accumulated belong jointly to the pair, but now must be divided between the two. They assign points to the items as follows: Use the adjusted winner procedure to determine a fair division of the property.

Question 5

Assume that Bob, Carol, Ted, and Alice view a cake as shown below: Notice that each player views the cake as having 16 units of value. Assume that the four players use the last-diminisher method to divide the cake among themselves. Give a step-by-step description of the cuts made and the pieces with which each player exits assuming that the order in which they proceed is Alice, then Ted, then Carol, and then Bob.

Question 6

Assume that Bob, Carol, Ted, and Alice view a cake as shown below: Notice that each player views the cake as having 16 units of value. Assume that the four players use the last-diminisher method to divide the cake among themselves. Give a step-by-step description of the cuts made and the pieces with which each player exits assuming that the order in which they proceed is Bob, then Carol, then Ted, and then Alice.

Question 7

Suppose that Bob and Carol want to take turns, without knowing about the other's preference list, to allocate several mathematics and statistics textbooks currently held jointly. Their ranked preferences are listed below: If Bob chooses first, what is the division of the objects?

Question 8

John, Ken, and Linda inherit a painting. Their monetary bids on the painting are $25,200, $21,600, and $18,000, respectively. If Linda had known of John and Ken's evaluations in advance, should she have changed her own evaluation?

Question 9

Suppose players 1, 2, and 3 view a cake as shown below and that all cuts are vertical. Illustrate the envy-free procedure for n = 3 (yielding an allocation of part of the cake) by following the steps below:
a) Provide a total of three drawings to show how each player views a division of the cake by player 1 into three pieces he or she considers to be the same size or value. Label the pieces A, B, and C.
b) Redraw the picture from player 2's view and illustrate the trimmings of piece B that he or she would do. Label the trimmed piece QUOTE and the actual trimmings T.
c) Indicate which piece each player would choose (and what he or she thinks its size is) if the players choose in the following order: player 3, player 2, player 1, according to the rules of the procedure.

Question 10

Describe the suitability criteria in arriving at a final ranking of those needing a kidney.

Question 11

Two people inherit a painting. If their monetary bids on the painting are $12,560 and $9750, what is the fair distribution arrived at by the Knaster inheritance procedure?

Question 12

John, Ken, and Linda inherit a painting. If their monetary bids on the painting are $25,200, $21,600, and $18,000, respectively, what is the fair distribution arrived at by the Knaster inheritance procedure?

Question 13

Suppose that Bob and Carol view a cake as shown below: Assume that Bob and Carol know how each other values the cake (and that neither is spiteful). Suppose they are to divide the cake using the rules of divide-and-choose but not necessarily the strategies. If Bob is the divider, where will he cut the cake and how many units of value will Bob and Carol think he or she is receiving?

Question 14

Suppose that Bob, Carol, and Ted view a cake as shown below: Assume that each player regards a piece as acceptable if and only if it is at least five square units of value. Assume all cuts are vertical. Provide a total of three drawings to show how each player views a division of the cake by Bob into three pieces he considers to be the same size or value. Now use the Steinhaus lone-divider method to get a proportional allocation, indicating who approves of which piece and how large a piece each player thinks he or she is receiving.

Question 15

Describe the waiting time criteria in arriving at a final ranking of those needing a kidney.

Question 16

Suppose that Bob, Carol, and Ted view a cake as shown below: Assume that each player regards a piece as acceptable if and only if it is at least five square units of value. Assume all cuts are vertical. Provide a total of three drawings to show how each player views a division of the cake, by Bob, into three pieces he or she considers to be the same size or value.

Question 17

A husband and wife are getting divorced. The property to be divided, and the relative value of the items to each, are as follows: Use the adjusted winner procedure to determine a fair division of the property.

Question 18

In recent labor-management negotiations, several issues were identified, and points assigned to them indicating relative importance to each side, as follows: Use the adjusted winner procedure to determine a fair resolution between labor and management.

Question 19

Suppose players 1, 2, and 3 view a cake as shown below, and that all cuts are vertical. Describe the steps in using the last-diminisher method to divide the cake. Assume players will use the order of player 1, then player 2, and then player 3 and that all pieces will be cut from the left side of the cake. Use drawings to clarify your description.

Question 20

A parent leaves a house, a farm, and a piece of property to be divided equally among four children who submit dollar bids on these objects as follows: What is the fair division arrived at by the Knaster inheritance procedure?

Question 21

Use the following information to answer Questions
Albert, Bella, Cosmo, and Dante inherit their grandmother's estate, which consists of a rare first- edition novel and a piano. The dollar values that each heir places on the items are as follows: Use the Knaster inheritance procedure to divide the property.
Who gets the piano?

Question 22

Suppose Chris and Terry need to make a fair division of a stereo they jointly purchased. Suppose also that both agree Chris is entitled to 2/3 of the stereo and Terry to 1/3. Chris and Terry place monetary bids on the value of the stereo of $500 and $450, respectively. Modify the Knaster inheritance procedure to make a fair division of the stereo.

Question 23

Three people must make a fair division of four objects. Using the Knaster inheritance procedure, the value each person places on the objects is shown below. What is the division of the objects?

Question 24

Apply the remainder of the Selfridge-Conway procedure to what was obtained in Question 20 by completing the steps below:
a) Draw a picture of T from each player's view.
b) The procedure calls for the player (other than player 1) who did not receive the trimmed piece to divide T into three pieces he or she believes to be the same size. Illustrate this division, and label the pieces X, Y, and Z.
c) Indicate which parts of T (and the sizes or values) the players will choose when they go in the order: player 2, player 1, player 3.

Question 25

Use the following information to answer Questions
Albert, Bella, Cosmo, and Dante inherit their grandmother's estate, which consists of a rare first- edition novel and a piano. The dollar values that each heir places on the items are as follows: Use the Knaster inheritance procedure to divide the property.
Who gets the novel?

Question 26

Suppose Tony and Jess want to take turns, using the bottom-up strategy, to allocate several items. If Tony and Jess rank the same item as first on their preference lists, who gets this item?

Question 27

Suppose Dave and Merrie want to take turns, using the bottom-up strategy, to allocate six treats. Their ranked preferences are listed below: If Dave chooses first, what is the division of the objects?

Question 28

Use the following information to answer Questions
Albert, Bella, Cosmo, and Dante inherit their grandmother's estate, which consists of a rare first- edition novel and a piano. The dollar values that each heir places on the items are as follows: Use the Knaster inheritance procedure to divide the property.
How much cash does Cosmo end up with?

Question 29

Suppose Tony and Jess want to take turns, using the bottom-up strategy, to allocate several items. If Tony and Jess have the same last choice on their preference lists, who gets this item?

Question 30

Use the following information to answer Questions
Albert, Bella, Cosmo, and Dante inherit their grandmother's estate, which consists of a rare first- edition novel and a piano. The dollar values that each heir places on the items are as follows: Use the Knaster inheritance procedure to divide the property.
How much value does Bella think she ends up with?

Question 31

Explain why the divide-and-choose method of cake division is considered an envy-free procedure for dividing a cake between two participants.

Question 32

Suppose Kelly and Jan want to take turns, using the bottom-up strategy, to allocate several textbooks currently held jointly. Their ranked preferences are listed below: If Kelly chooses first, how are books allocated?

Question 33

Tom and Jerry must rake all the leaves in a yard. They each view the yard as shown below and agree to divide the chore by the divide-and-choose method. Tom will divide the yard in two parts and Jerry will choose the side he has to rake. Describe the results of this fair division.

Question 34

Suppose Kelly and Jan want to take turns, using the bottom-up strategy, to allocate several textbooks currently held jointly. Their ranked preferences are listed below: If Jan chooses first, how are books allocated?

Question 35

Four people must make a fair division of two classic cars. They use the Knaster inheritance procedure. Their bids are shown below: What is the division of the objects?

Question 36

Use the following information to answer Questions
Albert, Bella, Cosmo, and Dante inherit their grandmother's estate, which consists of a rare first- edition novel and a piano. The dollar values that each heir places on the items are as follows: Use the Knaster inheritance procedure to divide the property.
How much cash does Dante end up with?

Question 37

Abby and Annie must make a fair division of four objects inherited from their grandmother. If Abby and Annie use the adjusted winner procedure, what is the division of the objects?

Question 38

If Jack and Jill want to take turns, using the bottom-up strategy, to allocate 10 items, under what situation could it be possible that each person would get their top five choices?

Question 39

Aunt Sally dies and leaves her estate to be shared among Ariel, Binky, and Chris. The estate consists of a painting and a sculpture by a modern artist. Aunt Sally had decreed that Ariel was to receive half her estate and that Binky and Chris were to share equally the remaining half. The values that each heir placed on the painting and sculpture are given below. Modify the Knaster inheritance procedure to make a fair division of the estate.

Question 40

Paul and Henry must make a fair division of three objects. If Paul and Henry use the adjusted winner procedure, what is the division of the objects?

Question 41

Use the following information to answer Questions
Jack and Jill went up the hill and found an antique water pail. Jack and Jill must now make a fair division of the pail using the Knaster inheritance procedure. Jack bids $80 for the value of the pail and Jill bids $65.
How much value does Jill think she ends with?

A) $36.25
B) $32.50
C) $40
D) $43.75

Question 42

Use the following information to answer Questions
Jack and Jill went up the hill and found an antique water pail. Jack and Jill must now make a fair division of the pail using the Knaster inheritance procedure. Jack bids $80 for the value of the pail and Jill bids $65.
How much value does Jack believe he ends with?

A) $36.25
B) $32.50
C) $40
D) $43.75

Question 43

Andi and Toni must make fair division of a car, a house, and a boat. The point values they place on the objects are given below. Using the adjusted winner procedure, what does Andi end with?

A) house and boat
B) house and 2/5 of boat
C) house and 2/7 of car
D) house and 3/5 of boat

Question 44

Chris and Terry must make a fair division of a stereo, television, and microwave. They place point values on the objects as shown below. Using the adjusted winner procedure, what point value does Chris feel he receives in the end?

A) 60
B) 62.14
C) 50
D) 37.86

Question 45

Use the following information to answer Questions
At the end of the lease on their apartment, Abe and Bonnie decide to rent separate places and so must make a fair division of the property they purchased together. The items to be divided and the point values each person places on them are given below.
If Abe and Bonnie use the adjusted winner procedure to divide the property, how many points of value will each person think they end up with?

A) 55.67
B) 76.67
C) 54.23
D) 56.50

Question 46

Use the following information to answer Questions
Tom and Sandy must make a fair division of three objects left by their great-aunt Sally. They have assigned points to the objects as shown below.
Using the adjusted winner procedure, who gets the car?

A) Tom
B) Sandy

Question 47

Andi, Chris, and Kim must make a fair division of a car left to them by their father. Using the Knaster inheritance procedure, the values they bid on the car are given below. Find the results of the fair division.
$\begin{array}{lccc} &\text { Andi}& \text { Chris}& \text { Kim }\\ \text { Car } &\$ 2400& \$ 3000& \$ 2700\end{array}$

A) Chris gets the car and pays $1900; Andi gets $900, and Kim gets $1000.
B) Chris gets the car, Andi gets $800; Kim gets $900.
C) Chris gets the car and pays $1500; Andi gets $1200, and Kim gets $1350.
D) Chris gets the car and pays $2550; Andi gets $1200, and Kim gets $1350.

Question 48

Andi, Chris, and Kim must make a fair division of a car left to them by their father. Using the Knaster inheritance procedure, the values they bid on the car are given below.
$\begin{array}{lccc} &\text { Andi}& \text { Chris}& \text { Kim }\\ \text { Car } &\$ 2400& \$ 3000& \$ 2700\end{array}$

How much value does Chris think he ends up with?

A) $3000
B) $1300
C) $1100
D) $1000

Question 49

Use the following information to answer Questions
At the end of the lease on their apartment, Toni and Terry decide to rent separate places and so must make a fair division of the property they purchased together. The items to be divided and the point values each person places on them are given below.
If Toni and Terry use the adjusted winner procedure to divide the property, what will Terry end with?

A) CD player, microwave, and TV
B) CD player, microwave, and 1/6 TV
C) CD player and microwave only
D) CD player, microwave, and 4/13 TV

Question 50

Use the following information to answer Questions
Tom and Sandy must make a fair division of three objects left by their great-aunt Sally. They have assigned points to the objects as shown below.
Using the adjusted winner procedure, what does Tom end up with?

A) car and painting
B) car only
C) painting and 1/5 car
D) car and 3/7 painting

Question 51

Use the following information to answer Questions
At the end of the lease on their apartment, Toni and Terry decide to rent separate places and so must make a fair division of the property they purchased together. The items to be divided and the point values each person places on them are given below.
If Toni and Terry use the adjusted winner procedure to divide the property, who gets the microwave?

A) Toni
B) Terry

Question 52

Use the following information to answer Questions
At the end of the lease on their apartment, Toni and Terry decide to rent separate places and so must make a fair division of the property they purchased together. The items to be divided and the point values each person places on them are given below.
If Toni and Terry use the adjusted winner procedure to divide the property, how many points of value will each person think they end up with?

A) 45
B) 50
C) 54.23
D) 56.5

Question 53

Which of the following fair-division procedures is not envy-free?

A) adjusted winner procedure
B) Knaster inheritance procedure
C) last-diminisher method
D) divide-and-choose

Question 54

Chris and Terry must make a fair division of a stereo, television, and microwave. They place point values on the objects as shown below. Using the adjusted winner procedure, what does Terry receive?

A) stereo and 1/4 of microwave
B) microwave and 6/7 of television
C) stereo and 6/7 of microwave
D) stereo and microwave

Question 55

Andi and Toni must make fair division of a car, a house, and a boat. The point values they place on the objects are given below. Using the adjusted winner procedure, what point value does Andi end with?

A) 65
B) 55
C) 50
D) 60

Question 56

Use the following information to answer Questions
Jack and Jill went up the hill and found an antique water pail. Jack and Jill must now make a fair division of the pail using the Knaster inheritance procedure. Jack bids $80 for the value of the pail and Jill bids $65.
What is the outcome of the fair division?

A) Jack gets the pail and pays Jill $36.25.
B) Jack gets the pail and pays Jill $40.
C) Jack gets the pail and pays Jill $32.50.
D) Jack gets the pail and Jill gets $65.

Question 57

Use the following information to answer Questions
At the end of the lease on their apartment, Abe and Bonnie decide to rent separate places and so must make a fair division of the property they purchased together. The items to be divided and the point values each person places on them are given below.
If Abe and Bonnie use the adjusted winner procedure to divide the property, who gets the bookcase?

A) Abe
B) Bonnie

Question 58

Define a Vickrey auction

Question 59

Use the following information to answer Questions
Tom and Sandy must make a fair division of three objects left by their great-aunt Sally. They have assigned points to the objects as shown below.
Using the adjusted winner procedure, how many points of value does Tom feel he ends with?

A) 67.14
B) 90
C) 50
D) 50.43

Question 60

Use the following information to answer Questions
At the end of the lease on their apartment, Abe and Bonnie decide to rent separate places and so must make a fair division of the property they purchased together. The items to be divided and the point values each person places on them are given below.
If Abe and Bonnie use the adjusted winner procedure to divide the property, what will Bonnie end with?

A) plant, picture, and 1/3 quilt
B) picture only
C) plant and picture only
D) plant, quilt, and 1/3 picture

Question 61

Use the following information to answer Questions
Three children must make fair division of a painting and sculpture left them by their mother. Using the Knaster inheritance procedure, the value each child places on the objects is shown below.
What does B get after the fair division?

A) painting and $366.67 b) painting and pays $3233.34
C) painting and pays $3600
D) painting and pays $2250

Question 62

Use the following information to answer Questions
Ashley, Brook, Chris, and Dana must make a fair division of the estate of their grandfather Wilbur. The estate consists of an old coin and a suit. Using the Knaster inheritance procedure, the value each person places on each item is shown below.
Who gets the suit?

A) Ashley
B) Brook
C) Chris
D) Dana

Question 63

What is the net value to Brook of her inheritance?

A)
B)
C)
D) $10

Question 64

Use the following information to answer Questions
Ashley, Brook, Chris, and Dana must make a fair division of the estate of their uncle Bob. The estate consists of a farm house, a car, and a horse. Using the Knaster inheritance procedure, the value each person places on each item is shown below.
Who gets the horse?

A) Ashley
B) Brook
C) Chris
D) Dana

Question 65

Peyton Young's synopsis of the fair division procedure for kidney allocation adopted by OPTN has several main criteria to arrive at a final ranking of those needing a kidney. Which criterium deals with the percent sensitized?

A) waiting time
B) suitability
C) disadvantage
D) None of the above.

Question 66

Use the following information to answer Questions
Suppose that there are five potential kidney recipients with the following characteristics:
What are the total points awarded to the potential recipient A?

A) 10
B) 15
C) 17
D) 18

Question 67

Use the following information to answer Questions
Ashley, Brook, Chris, and Dana must make a fair division of the estate of their grandfather Wilbur. The estate consists of an old coin and a suit. Using the Knaster inheritance procedure, the value each person places on each item is shown below.
Who gets the coin?

A) Ashley
B) Brook
C) Chris
D) Dana

Question 68

What does Brook think the inheritance is worth?

A)
B)
C)
D) $10

Question 69

Use the following information to answer Questions
Ashley, Brook, Chris, and Dana must make a fair division of the estate of their uncle Bob. The estate consists of a farm house, a car, and a horse. Using the Knaster inheritance procedure, the value each person places on each item is shown below.
What does Ashley end the fair division with?

A) car and $18,000 b) car and $8000
C) car and $22,000
D) car and $9731.25

Question 70

Use the following information to answer Questions
Suppose that there are five potential kidney recipients with the following characteristics:
What are the total points awarded to the potential recipient E?

A) 10
B) 15
C) 17
D) 18

Question 71

What does Brook have at the end of the process?

A)
B) the coin plus
C) the coin minus
D) the coin with no cash gained or spent

Question 72

What does Ashley have at the end of the process?

A)
B)
C)
D)

Question 73

Use the following information to answer Questions
Ashley, Brook, Chris, and Dana must make a fair division of the estate of their uncle Bob. The estate consists of a farm house, a car, and a horse. Using the Knaster inheritance procedure, the value each person places on each item is shown below.
What value does Chris end the fair division with?

A) $9731
B) $14,250
C) $15,981.25
D) $1731.25

Question 74

What does Dana have at the end of the process?

A)
B)
C)
D)

Question 75

Use the following information to answer Questions
Three children must make fair division of a painting and sculpture left them by their mother. Using the Knaster inheritance procedure, the value each child places on the objects is shown below.
Who gets the sculpture?

A) A
B) B
C) C

Question 76

What does Chris have at the end of the process?

A)
B) the suit minus
C) the suit plus
D) ttthe suit with no cash gained or spent

Question 77

Does Ashley get more than one quarter of what she thinks the inheritance is worth?

A) Yes
B) No

Question 78

Does Brook get more than one quarter of what she thinks the inheritance is worth?

A) Yes
B) No

Question 79

Use the following information to answer Questions
Three children must make fair division of a painting and sculpture left them by their mother. Using the Knaster inheritance procedure, the value each child places on the objects is shown below.
What does A get after the fair division?

A) $2466.67
B) $2100
C) sculpture and $166.67
D) $3150

Question 80

What does Ashley think the inheritance is worth?

A)
B)
C)
D) $10