Fair Division Assignment Modern Math These questions are designed to help you practice the fair division topics from class. You may want to do work on a separate piece of paper. 1. The application of any fair division method requires certain assumptions. For example, the success of the method of sealed bids requires that each heir place a value on each object in the estate. If any heir considers an object priceless, the method will fail. List at least one other assumption that is necessary for the method of sealed bids to work and explain why it's necessary. 2. Suppose that Garfield and Marmaduke are heirs to an estate that contains only a house. They decide to use the method of sealed bids, where Garfield bids $70,000 and Marmaduke bids $60,000. a) What does Garfield feel is a fair share? How about Marmaduke? b) Who will receive the house in the initial allotment? c) How much of a surplus will the estate receive? d) Describe the final allotment in this settlement. 3. Amy, Bender, Leela, and Scruffy are heirs to an estate that consists of a silk-screen of Elvis painting, a \"suped-up\" motor boat, and a prize winning cow named Bootsy. They submit the bids below. Use the method of sealed bids to completely divide the estate between the heirs. Elvis Painting Motor Boat Bootsy, the Cow 4. Amy $6,000 $8,000 $400 Bender $3,000 $10,000 $200 Leela $2,000 $9,000 $1,000 Scruffy $4,000 $11,000 $900 Describe the difference between a discrete fair division problem and a continuous fair division problem. Give examples of each. 5. Logan and Victor want to divide a cake using the divider-chooser method. They draw straws and determine that Logan will be the divider and Victor will be the chooser. a) Assuming each player plays the game fairly, is it possible for Logan to believe that his share is worth 60% of the cake? Why or why not? b) Is it possible for Victor to believe her share is worth 60% of the cake? Why or why not? 6. Five investors purchase a piece of land for one million dollars and decide to divide it fairly among themselves using the last diminisher method. The investors are Bob, Patrick, Sandy, Gary, and Plankton (in that order.) In round 1, the only diminisher is Patrick. In round 2, the only diminisher is Gary. In round 3, everyone diminishes. Describe who gets the land at the end of each round, as well as what happens in the final round. 7. Three players are going to divide a parcel of land using the lone divider method. The choosers both bid on only s3. Is it possible for the divider to end up with any part of s3? How should this problem be settled? Explain the final division thoroughly. 8. Francis, Lewis, Bill, and Zoey agree to fairly divide a cake using the lone divider method. The table below shows how each player values each of the four slices that have been cut by the divider as percentages of the whole. Francis Lewis Bill Zoey s1 25% 20% 20% 30% s2 25% 32% 22% 32% s3 25% 28% 45% 28% s4 25% 20% 13% 10% a) Assuming all players player honestly, which player was the divider? b) Assuming that he plays fairly, what should be Zoey's declaration? c) Describe a fair division of the cake