Exam 13: Fair Division

Use the following information to answer the 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: Novel 10,000 6,000 Piano 2,000 5,000 500 100 4,000 100 Use the Knaster inheritance procedure to divide the property. -How much value does Bella think she ends up with?

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. Item Ashley Brook Chris Dana Coin \ 4 \ 6 \ 4 \ 4 Suit \ 4 \ 4 \ 7 \ 4 -Who gets the coin?

Kim and Tori must make a fair division of three objects. Using the adjusted winner procedure, what does Kim end up with? Object Kim's points Tori's points Ring 20 35 Necklace 70 45 Bracelet 10 20

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.

Define a Vickrey auction

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 many units will Toni believe he gets?

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 $B ^ { \prime }$ 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.

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. Andi Chris Kim Car \ 2400 \ 3000 \ 2700 How much value does Chris think he ends up with?

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. Will Betty trim the piece before handing it to Chris?

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. -Who will make the last cut and thus keep the first piece of cake?

Use the following information to answer the 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. Item Toni's points Terry's points CD player 15 20 Sofa 30 25 Microwave 20 25 TV 35 30 -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?

Use the following information to answer the 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. Item Ashley Brook Chris Dana House \ 35,000 \ 50,000 \ 48,000 \ 42,000 Car \ 3000 \ 2500 \ 2750 \ 2800 Horse \ 6000 \ 5800 \ 6250 \ 5200 -What value does Chris end the fair division with?

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

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.

Use the following information to answer the Questions: In a Vickrey auction, the three highest bids (by bidders A, B, and C) are $100, $101, and $500. -Who wins the auction?

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.

Use the following information to answer the 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. Item Ashley Brook Chris Dana House \ 35,000 \ 50,000 \ 48,000 \ 42,000 Car \ 3000 \ 2500 \ 2750 \ 2800 Horse \ 6000 \ 5800 \ 6250 \ 5200 -Who gets the horse?

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, which side will Toni choose?

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 ecology book chosen?

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. Object Ariel Binky Chris Painting 4,000 6,300 6,000 Sculpture 2,300 1,800 2,400