2 An Activity Think about how you would fairly divide the following: The last piece of cake with your brothers or sistersYour parents’ estate between all their childrenThe seats on the student council if it is composed of 20 members divided among 3 classes: 332 sophomores, 288 juniors, and 279 seniors.
3 Estate DivisionA fair division problem can be either discrete or continuous.Dividing a house among heirs and dividing seats among student council are examples of discrete case.Discrete divisions occur whenever objects cannot be meaningfully separated into pieces.Dividing a cake is continuous.
4 Fair DivisionToday we will look at how estates are divided fairly between heirs.We will use an algorithm for dividing an estate among heirs, that produces an interesting paradox.
5 Estate Division Algorithm Each heir submits a bid for each item in the estate. Bids are not made on cash in the estate because it is a continuous medium that can be divided equally without controversy.A fair share is determined for each heir by finding the sum of his or her bids and dividing this sum, by the number of heirs.
6 Estate Division Algorithm (cont’d) Each item in the estate is given to the heir who bid the highest on that item.Each heir is given an amount of cash from the estate that is equal to his or her share (from Step 2) less the amount that the heir bid on the objects he or she received. If this amount is negative, the heir pays that amount into the estate.The remaining cash in the estate is divided equally among the heirs.
7 ExampleAmanda, Brian and Charlene are heirs to an estate that includes a house, a boat, a car, and $75,000 in cash. Each submits a bid for the house, boat and car. The bids are summarized in the following table.HouseBoatCarAmanda$40, 000$3, 000$5, 000Brian$35, 000$8, 000Charlene$38, 000$4, 000$9, 000
8 Example (cont’d)The entries in Amanda’s row, for example, indicate the value to Amanda of each item in the estate.A fair share is determined for each heir.Amanda: ($40,000 + $3,000 + $5,000 +$75,000)/3 =$41,000Brian: ($35,000 + $5,000 + $8,000 +$75,000)/3 =$41,000Charlene: ($38,000 + $4,000 + $9,000 +$75,000)/3 =$42,000
9 Example (cont’d)The house is given to Amanda, the boat to Brian and the car to Charlene. Cash equal to the difference between the fair share and the value of the awarded items is given to each heir.Amanda: $41,000-$40,000=$1,000Brian: $41,000-$5,000=$36,000Charlene: $42,000-$9,000=$33,000
10 Example (cont’d)The cash given to the heirs totals $70,000 and the remaining $5,000 is divided equally.Thus each heir receives $ , more than a fair share.Notice that each person receives more than a fair share of the estate.
11 Fair Division Algorithm The application of any fair division algorithm requires certain assumptions, or axioms. For example, the success of the estate division algorithm requires that each heir place a value on each object in the estate. If any heir considers an object priceless, the algorithm will fail.
12 Practice ProblemsSuppose that Garfield and Marmaduke are heirs to an estate that contains only a house. Garfield bids $70,000 and Marmaduke bids $60,000.a. What does Garfield feel is a fair share? Marmaduke?b. What is the difference between Garfield’s fair share and Garfield’s bid for the house?
13 Problems (cont’d)Because the value of the house is more than Garfield’s fair share, Garfield must pay cash to the estate. How much cash must Garfield pay?Marmaduke is given an amount of cash from Garfield’s payment equal to Marmaduke’s fair share. How much does Marmaduke receive? If the remaining cash is divided equally, what will be the final value of Marmaduke’s settlement? Of Garfield’s?
14 Problems (cont’d)e. If the division between Garfield and Marmaduke were settled in the usual way, Marmaduke would be given half of Garfield’s bid. Compare the final settlements for Garfield and Marmaduke if this method is used with the settlement in part d. Which result do you think is best?
15 Problems (cont’d)Amy, Bart and Carol are heirs to an estate that consists of a valuable painting, a car, a New York Yankees season ticket and $5,000 in cash. They submit the bids as shown in the table below.PaintingCarTicketAmy$2,000$4,000$500Bart$5,000$100Carol$3,000$300
16 Problems (cont’d)Use the method of this lesson to divide the estate among the heirs. For each heir, state their fair share, the items received, the amount of cash, and the final settlement.Hint: When a division requires hat some heirs pay into the estate, be sure to keep an accurate record of the estate’s cash.
17 Problems (cont’d)Suppose two heirs submit an identical highest bid for an item. How do you think you would resolve this?
18 Problems (cont’d)Alan, Betty and Carl are heirs to an estate and have submitted the bids shown below.HouseBoatCarAlan$55,000$3,000$8,000Betty$60,000$4,000$6,000Carl$56,000$7,000
19 Problems (cont’d)The awarding of items in the estate can be indicated in a matrix, as shown at the right.The numbers in the first column indicate the items that Alan received, where the 1st row is the house, the 2nd row is the boat and the 3rd row is the car.
20 Problems (cont’d)A new matrix can be computed by writing the table, as a matrix, alongside the first as shown below
21 Problems (cont’d)The new matrix is computed by multiplying the two matrices.Calculate the new matrix.The $8,000 in the first row and first column can be interpreted as the value to Alan of the items he received. What do all the other entries mean?