Fair DivisionFair Division

Estate DivisionEstate Division

An ActivityAn Activity

Think about how you would fairly divide Think about how you would fairly divide the following:the following:The last piece of cake with your brothers or The last piece of cake with your brothers or

sisterssistersYour parents’ estate between all their childrenYour parents’ estate between all their childrenThe seats on the student council if it is The seats on the student council if it is

composed of 20 members divided among 3 composed of 20 members divided among 3 classes: 332 sophomores, 288 juniors, and classes: 332 sophomores, 288 juniors, and 279 seniors.279 seniors.

Estate DivisionEstate Division

A fair division problem can be either A fair division problem can be either discretediscrete or or continuouscontinuous. .

Dividing a house among heirs and dividing Dividing a house among heirs and dividing seats among student council are examples seats among student council are examples of discrete case.of discrete case.

Discrete divisions occur whenever objects Discrete divisions occur whenever objects cannot be meaningfully separated into cannot be meaningfully separated into pieces.pieces.

Dividing a cake is continuous.Dividing a cake is continuous.

Fair DivisionFair Division

Today we will look at how estates are Today we will look at how estates are divided fairly between heirs.divided fairly between heirs.

We will use an algorithm for dividing an We will use an algorithm for dividing an estate among heirs, that produces an estate among heirs, that produces an interesting paradox.interesting paradox.

Estate Division AlgorithmEstate Division Algorithm

1.1. Each heir submits a bid for each item in Each heir submits a bid for each item in the estate. Bids are not made on cash in the estate. Bids are not made on cash in the estate because it is a continuous the estate because it is a continuous medium that can be divided equally medium that can be divided equally without controversy.without controversy.

2.2. A fair share is determined for each heir A fair share is determined for each heir by finding the sum of his or her bids and by finding the sum of his or her bids and dividing this sum, by the number of heirs.dividing this sum, by the number of heirs.

Estate Division Algorithm (cont’d)Estate Division Algorithm (cont’d)

3.3. Each item in the estate is given to the heir Each item in the estate is given to the heir who bid the highest on that item.who bid the highest on that item.

4.4. Each heir is given an amount of cash from Each heir is given an amount of cash from the estate that is equal to his or her share the estate that is equal to his or her share (from Step 2) less the amount that the heir (from Step 2) less the amount that the heir bid on the objects he or she received. If this bid on the objects he or she received. If this amount is negative, the heir pays that amount is negative, the heir pays that amount into the estate.amount into the estate.

5.5. The remaining cash in the estate is divided The remaining cash in the estate is divided equally among the heirs.equally among the heirs.

ExampleExample

Amanda, Brian and Charlene are heirs to Amanda, Brian and Charlene are heirs to an estate that includes a house, a boat, a an estate that includes a house, a boat, a car, and $75,000 in cash. Each submits a car, and $75,000 in cash. Each submits a bid for the house, boat and car. The bids bid for the house, boat and car. The bids are summarized in the following table.are summarized in the following table.

HouseHouse BoatBoat CarCar

AmandaAmanda $40, 000$40, 000 $3, 000$3, 000 $5, 000$5, 000

BrianBrian $35, 000$35, 000 $5, 000$5, 000 $8, 000$8, 000

CharleneCharlene $38, 000$38, 000 $4, 000$4, 000 $9, 000$9, 000

Example (cont’d)Example (cont’d)

The entries in Amanda’s row, for example, The entries in Amanda’s row, for example, indicate the value to Amanda of each item in indicate the value to Amanda of each item in the estate.the estate.

A fair share is determined for each heir.A fair share is determined for each heir.

Amanda: ($40,000 + $3,000 + $5,000 +$75,000)/3 =$41,000Amanda: ($40,000 + $3,000 + $5,000 +$75,000)/3 =$41,000

Brian: ($35,000 + $5,000 + $8,000 +$75,000)/3 =$41,000Brian: ($35,000 + $5,000 + $8,000 +$75,000)/3 =$41,000

Charlene: ($38,000 + $4,000 + $9,000 +$75,000)/3 Charlene: ($38,000 + $4,000 + $9,000 +$75,000)/3 =$42,000=$42,000

Example (cont’d)Example (cont’d)

The house is given to Amanda, the boat to The house is given to Amanda, the boat to Brian and the car to Charlene. Cash equal Brian and the car to Charlene. Cash equal to the difference between the fair share to the difference between the fair share and the value of the awarded items is and the value of the awarded items is given to each heir.given to each heir.

Amanda: $41,000-$40,000=$1,000Amanda: $41,000-$40,000=$1,000

Brian: $41,000-$5,000=$36,000Brian: $41,000-$5,000=$36,000

Charlene: $42,000-$9,000=$33,000Charlene: $42,000-$9,000=$33,000

Example (cont’d)Example (cont’d)

The cash given to the heirs totals $70,000 The cash given to the heirs totals $70,000 and the remaining $5,000 is divided and the remaining $5,000 is divided equally. equally.

Thus each heir receives $1666.67, more Thus each heir receives $1666.67, more than a fair share.than a fair share.

Notice that each person receives more Notice that each person receives more than a fair share of the estate.than a fair share of the estate.

Fair Division AlgorithmFair Division Algorithm

The application of any fair division The application of any fair division algorithm requires certain assumptions, algorithm requires certain assumptions, or axioms. For example, the success of or axioms. For example, the success of the estate division algorithm requires that the estate division algorithm requires that each heir place a value on each object in each heir place a value on each object in the estate. If any heir considers an the estate. If any heir considers an object priceless, the algorithm will fail.object priceless, the algorithm will fail.

Practice ProblemsPractice Problems

1.1. Suppose that Garfield and Marmaduke Suppose that Garfield and Marmaduke are heirs to an estate that contains only a are heirs to an estate that contains only a house. Garfield bids $70,000 and house. Garfield bids $70,000 and Marmaduke bids $60,000.Marmaduke bids $60,000.a. What does Garfield feel is a fair a. What does Garfield feel is a fair share? Marmaduke?share? Marmaduke?b. What is the difference between b. What is the difference between Garfield’s fair share and Garfield’s bid Garfield’s fair share and Garfield’s bid for the house?for the house?

Problems (cont’d)Problems (cont’d)

c.c. Because the value of the house is more than Because the value of the house is more than Garfield’s fair share, Garfield must pay cash to Garfield’s fair share, Garfield must pay cash to the estate. How much cash must Garfield the estate. How much cash must Garfield pay?pay?

d.d. Marmaduke is given an amount of cash from Marmaduke is given an amount of cash from Garfield’s payment equal to Marmaduke’s fair Garfield’s payment equal to Marmaduke’s fair share. How much does Marmaduke receive? share. How much does Marmaduke receive? If the remaining cash is divided equally, what If the remaining cash is divided equally, what will be the final value of Marmaduke’s will be the final value of Marmaduke’s settlement? Of Garfield’s?settlement? Of Garfield’s?

Problems (cont’d)Problems (cont’d)

e. If the division between Garfield and e. If the division between Garfield and Marmaduke were settled in the usual way, Marmaduke were settled in the usual way, Marmaduke would be given half of Marmaduke would be given half of Garfield’s bid. Compare the final Garfield’s bid. Compare the final settlements for Garfield and Marmaduke if settlements for Garfield and Marmaduke if this method is used with the settlement in this method is used with the settlement in part d. Which result do you think is best?part d. Which result do you think is best?

Problems (cont’d)Problems (cont’d)

2.2. Amy, Bart and Carol are heirs to an estate Amy, Bart and Carol are heirs to an estate that consists of a valuable painting, a car, a that consists of a valuable painting, a car, a New York Yankees season ticket and New York Yankees season ticket and $5,000 in cash. They submit the bids as $5,000 in cash. They submit the bids as shown in the table below. shown in the table below.

PaintingPainting CarCar TicketTicket

AmyAmy $2,000$2,000 $4,000$4,000 $500$500

BartBart $5,000$5,000 $2,000$2,000 $100$100

CarolCarol $3,000$3,000 $3,000$3,000 $300$300

Problems (cont’d)Problems (cont’d)

Use the method of this lesson to divide the Use the method of this lesson to divide the estate among the heirs. For each heir, estate among the heirs. For each heir, state their fair share, the items received, state their fair share, the items received, the amount of cash, and the final the amount of cash, and the final settlement.settlement.

Hint: When a division requires hat some Hint: When a division requires hat some heirs pay into the estate, be sure to keep heirs pay into the estate, be sure to keep an accurate record of the estate’s cash.an accurate record of the estate’s cash.

Problems (cont’d)Problems (cont’d)

3.3. Suppose two heirs submit an identical Suppose two heirs submit an identical highest bid for an item. How do you highest bid for an item. How do you think you would resolve this?think you would resolve this?

Problems (cont’d)Problems (cont’d)

3.3. Alan, Betty and Carl are heirs to an Alan, Betty and Carl are heirs to an estate and have submitted the bids estate and have submitted the bids shown below.shown below.

HouseHouse BoatBoat CarCar

AlanAlan $55,000$55,000 $3,000$3,000 $8,000$8,000

BettyBetty $60,000$60,000 $4,000$4,000 $6,000$6,000

CarlCarl $56,000$56,000 $3,000$3,000 $7,000$7,000

Problems (cont’d)Problems (cont’d)

The awarding of items in The awarding of items in the estate can be the estate can be indicated in a matrix, as indicated in a matrix, as shown at the right.shown at the right.

The numbers in the first The numbers in the first column indicate the column indicate the items that Alan items that Alan received, where the 1received, where the 1stst row is the house, the 2row is the house, the 2ndnd row is the boat and the row is the boat and the 33rdrd row is the car. row is the car.

0 1 0

0 1 0

1 0 0

L

NMMM

O

QPPP

Problems (cont’d)Problems (cont’d)

A new matrix can be computed by writing the A new matrix can be computed by writing the table, as a matrix, alongside the first as table, as a matrix, alongside the first as shown belowshown below

$55, $3, $8,

$60, $4, $6,

$56, $3, $7,

000 000 000

000 000 000

000 000 000

0 1 0

0 1 0

1 0 0

L

NMMM

O

QPPPL

NMMM

O

QPPP

Problems (cont’d)Problems (cont’d)

The new matrix is computed by multiplying The new matrix is computed by multiplying the two matrices.the two matrices.

a.a. Calculate the new matrix.Calculate the new matrix.

b.b. The $8,000 in the first row and first The $8,000 in the first row and first column can be interpreted as the value column can be interpreted as the value to Alan of the items he received. What to Alan of the items he received. What do all the other entries mean?do all the other entries mean?