Exam 13: Fair Division

Use the following information to answer the Questions: Tom and Huck need to paint a fence and agree to use a modification of the divide-and-choose procedure to make a fair division of the chore. The boys' views of the fence are shown below. (Remember, each boy wants to end up with the smallest possible amount to paint.) -Andi and Terry need to make a fair division of a sofa they bought when sharing an apartment. They agree that Andi is entitled to two thirds of the sofa and Terry is entitled to one third of the sofa. They agree to use a modification of the Knaster inheritance procedure to divide the sofa. Andi believes the sofa is worth $360 and Terry believes the sofa is worth $315. Using the Knaster inheritance procedure, how much will Andi pay Terry for the sofa?

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.

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

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: 1 st 2 nd 3rd 4 th 5 th Bob precalculus calculus algebra statistics trigonometry analysis Carol trigonometry algebra precalculus statistics calculus analysis If Bob chooses first, what is the division of the objects?

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

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

Use the following information to answer the Questions: Alex, Betty, and Chris each view a cake as shown below. They will share the cake using the last-diminisher method with the order Alex, Betty, and Chris cutting and passing off the right side of the cake. -After Alex cuts the cake, he will hand the portion on the right to Betty, who decides whether to trim the piece and then hands it off to Chris. Will Chris trim the piece he receives from Betty?

Use the following information to answer the Questions: Tom and Huck need to paint a fence and agree to use a modification of the divide-and-choose procedure to make a fair division of the chore. The boys' views of the fence are shown below. (Remember, each boy wants to end up with the smallest possible amount to paint.) -If Tom divides and Huck chooses, how many units of fence will Tom think each boy is painting?

Use the following information to answer the Questions: Suppose Kim and Chris want to take turns, using the bottom-up strategy, to allocate several textbooks currently held jointly. Their ranked preferences are listed below: Chris botany sociology physiology biology ecology chemistry Kim biology chemistry botany sociology ecology physiology -If Kim chooses first, when is the physiology book chosen?

Use the following information to answer the Questions: Alex, Betty, and Chris each view a cake as shown below. They will share the cake using the last-diminisher method with the order Alex, Betty, and Chris cutting and passing off the right side of the cake. -After Alex cuts the cake, he will hand the portion on the right to Betty. Where will Alex make his cut?

Use the following information to answer the 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. Object A B C Painting \ 4000 \ 6300 \ 6000 Sculpture \ 2300 \ 1800 \ 2400 -What does A get after the fair division?

What does Brook have at the end of the process?

Use the following information to answer the 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?

Use the following information to answer Questions Suppose Chris and Toni each view a cake as shown below. They agree to use the divide-and-choose procedure. -If Chris divides and Toni chooses, how much will Chris feel is left?

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? Object Chris's points Terry's point Stereo 25 45 Television 60 35 Microwave 15 20

Use the following information to answer the Questions: Suppose that there are five potential kidney recipients with the following characteristics: Potential Recipient Months Waiting Antigens Matched Percent Sensitized A 5 2 10 B 4 3 30 C 3.5 1 20 D 3 3 70 E 2 4 80 -What are the total points awarded to the potential recipient E?

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: 1 st 2 nd 3 rd 4th 5 th 6 th Kelly Jan painting design music architecture architecture painting sculpture sculpture design music textiles textiles If Kelly chooses first, how are books allocated?

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? Object Andi's points Toni's points Car 35 45 House 40 30 Boat 25 25

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.

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.