Lesson 2 - Weeblygvmath.weebly.com/uploads/5/6/5/3/56539921/2.2_estate_division.pdf · Lesson 2.2 • Estate Division 65 Exercises 1. The success of a fair division model requires

Lesson 2.2

Estate Division

A fair division problem can be either discrete or continuous. The problem ofdividing a house among heirs and that of dividing a student councilamong classes are examples of the discrete case. Discrete division occurswhenever the objects of the division cannot be meaningfully separatedinto pieces. Dividing a cake is an example of the continuous case becausethe cake can be divided into any number of pieces.

This lesson considers fair division of an estate among heirs. In yourdiscussions in Lesson 2.1, you may have decided that fairness is difficult todefine in some situations because different people place different valueson the same object. However, it is sometimes possible to use suchdifferences of opinion to the advantage of all those involved. The estatedivision model on the next page is an algorith that produces an appealingparadox: Each of the heirs receives a share that is larger than he or shethinks is fair.

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62 Chapter 2 • Fair Division

The first four steps of thealgorithm give each heirgoods and cash whose totalvalue equals what the heirfeels is a fair share of theestate. Extra cash awarded instep 5 is a bonus.

Whenever you encountera new model, it is helpful togive it a try. After consideringthe following example, youwill have the opportunity touse the model in this lesson’sexercises.

An Algorithm for Dividing an Estate1. Each heir submits a bid for each item in the estate. (Bids are not

made on cash in the estate because it can be divided equallywithout controversy.)

2. 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. (If there is cash in the estate, its total is included in each heir’s sum.)

3. Each item in the estate is given to the heir who bid the higheston it.

4. Each heir is given an amount of cash from the estate that isequal to his or her fair share (from step 2) less the amount theheir bid on the objects he or she receives. If this amount isnegative, the heir pays that amount into the estate’s cash.

5. Any remaining cash in the estate is divided equally among the heirs.

Wellington's Millions

mlive.comMay 24, 2011

Trustees for the $110million estate ofWellington R. Burt, the19th-century lumber baronwho froze out family fromhis fortunes for nearly acentury, distributed themillions of dollars to 12Burt heirs by a Mondaydeadline.

The estate’s distributioncloses the book on Burt’sstoried estate, which firstmade headlines when thelumber baron died in 1919.That’s when the public

learned his will featuredan odd stipulation freezingthe bulk of his fortunesfrom family until 21 yearsafter his last livinggrandchild’s death. His lastliving grandchild died in1989.

This month, the dozenheirs ironed out asettlement. They haveBurt’s blood in common,but many share little else.The youngest is 19; theoldest, 94. They’rescattered across eightstates—as far west asCalifornia and as far eastas Connecticut. One residesin Michigan.


63Lesson 2.2 • Estate Division

Estate Division ExampleAmanda, Brian, and Charlene are heirs to an estate that includes ahouse, a boat, a car, and $150,000 in cash.

Step 1Each heir submits bids for the house, boat, and car. The bids aresummarized in the following table, or matrix.

House Boat Car

Amanda $80,000 $5,000 $8,000Brian $70,000 $9,000 $11,000Charlene $76,000 $7,000 $13,000

For example, the entries in Amanda’s row indicate the value toAmanda of each item in the estate.

Step 2A fair share is determined for each heir.

Amanda: ($80,000 + $5,000 + $8,000 + $150,000) ÷ 3 = $81,000

Brian: ($70,000 + $9,000 + $11,000 + $150,000) ÷ 3 = $80,000

Charlene: ($76,000 + $7,000 + $13,000 + $150,000) ÷ 3 = $82,000

Step 3The house is given to Amanda, the boat toBrian, and the car to Charlene.

Step 4Each heir receives cash equal to thedifference between the fair share and thevalue of the awarded items.

Amanda: $81,000 – $80,000 = $1,000

Brian: $80,000 – $9,000 = $71,000

Charlene: $82,000 – $13,000 = $69,000

The graphing calculator featuresused to do Borda counts can beused to determine fair shares.Replay the calculation for the firstheir and edit it by moving thecursor and typing new digits whennecessary. Delete and Insertfeatures can be used when digitsmust be eliminated or introduced.

Technology Note



Step 5The cash that is given to the heirs totals $141,000, which leaves $150,000 – $141,000 = $9,000 in the estate. Each heir receives a bonus of$9,000 ÷ 3 = $3,000.

This example’s results can be summarized in a matrix:

Amanda Brian Charlene

Total of bids and cash $243,000 $240,000 $246,000Fair share $81,000 $80,000 $82,000Items received House Boat CarValue of items received $80,000 $9,000 $13,000Initial cash received $1,000 $71,000 $69,000Share of remaining cash $3,000 $3,000 $3,000

For each heir, totaling the last three rows of the matrix gives thevalue the heir attaches to the items and cash received. For example,Amanda feels the value of her share of the estate is $80,000 + $1,000 +$3,000 = $84,000, which is more than the $81,000 that Amanda feels is afair share.

The final settlements for each heir are:

Amanda: the house and $4,000

Brian: the boat and $74,000

Charlene: the car and $72,000

FOR BETTER OR WORSE © 1992 Lynn Johnston Productions. Dist. By Universal Press Syndicate.Reprinted with permission. All rights reserved.



Exercises1. The success of a fair division model requires certain assumptions, or

axioms. For example, this lesson’s algorithm works successfully onlyif each heir is capable of placing a value on each of the estate’sobjects. If any heir considers an object priceless or is otherwiseincapable of placing a dollar value on an object, the algorithmfails. State at least one other axiom that you think is necessary forthe success of this algorithm.

2. Calvin and Hobbes are using this lesson’s algorithm to divide anestate composed of a wagon, a sled, and $20 in cash. Calvin bids$20 for the wagon and $10 for the sled. Hobbes bids $10 for thewagon and $15 for the sled.

a. What does Calvin think is a fair share of the estate? Hobbes?

b. What value does Calvin place on his final settlement? Hobbes?Explain.

James Brown's Heirs Settle Estate Feud

IMDb.comMay 27, 2009

James Brown's heirs haveended their bitter feudover his estate. A SouthCarolina judge has signedoff on a settlementreached by the Godfatherof Soul's wife andchildren.

His wife, Tomi Rae Hynie,and six adult childrenhave fought over his richessince his death in 2006.His children were namedin his will, but his wife

and their son JamesBrown II were not. Thewill was signed beforetheir marriage and beforeJames II was born.

Under the terms of thesettlement, half thelegend's assets will go tohis charitable trust to payfor his grandchildren'seducation and help poorchildren in South Carolinaand Georgia, a quarterwill go to Hynie andJames II, and the rest tohis six children.

James Brown



3. Garfield and Marmaduke are heirs to an estate that contains only asummer cottage. Garfield bids $80,000. Marmaduke bids $70,000.

a. What does Garfield feel is a fair share? Marmaduke?

b. What is the difference between Garfield’s bid and Garfield’s fairshare?

c. Because the value Garfield assigned to the cottage is more thanGarfield’s fair share, Garfield must pay cash into the estate. Howmuch cash must Garfield pay?

d. Marmaduke is given an amount of cash from Garfield’s paymentequal to Marmaduke’s fair share. How much does Marmadukereceive? If the remaining cash is divided equally, what is thefinal value of Marmaduke’s settlement? Of Garfield’s settlement?

e. Garfield must borrow money in order to pay into the estate, andthe interest on this loan is $2,000. Do you think this should beconsidered when arriving at a settlement? If so, suggest how thesettlement can be revised.

f. If the division between Garfield and Marmaduke is settled byanother model that is frequently used to divide estates,Marmaduke receives half of Garfield’s bid. Compare the finalsettlements for Garfield and Marmaduke by this model with thesettlements in part d. Of the two results, which do you think isfairest? Explain.

4. Amy, Bart, and Carol are heirs to an estate that consists of avaluable painting, a motorcycle, a World Series ticket, and $5,000in cash. Their bids are shown in the following matrix.

Painting Motorcycle Ticket

Amy $2,000 $4,000 $500Bart $5,000 $2,000 $100Carol $3,000 $3,000 $300

a. Use this lesson’s algorithm to divide the estate among the heirs.For each heir, state the fair share, the items received, theamount of cash, and the final settlement. Summarize yourresults in a matrix.



A helpful tip: It is relatively easy to lose track of the estate’s cash ifseveral payments are made into and out of the estate. Errors can be avoided by tracking the cash with a small table designed forthat purpose:

Cash in the estate $5,000Received from Amy ______Received from Bart ______Paid to Carol ______Cash remaining ______

b. It is common for one or more heirs to pay into an estate. Thislesson’s estate division model fails if an heir who must pay intothe estate cannot do so. Suggest a way to modify the model forsituations in which one or more heirs cannot raise the cashneeded to complete the division.

5. Suppose that in the division of Exercise 4, Amy received previousfinancial support from the estate in the form of a loan to paycollege tuition. A will states that she is to receive only 20% of theestate, whereas Bart and Carol are to receive 40% each. Adapt themodel of this lesson to this situation and describe a fair division ofthe estate.

6. If two heirs submit identical highest bids for an item, how wouldyou resolve the tie?

7. Alan, Betty, and Carl are heirs to an estate. They submit the bidsshown in the following table.

House Boat Car

Alan $115,000 $6,000 $13,000Betty $120,000 $7,000 $11,000Carl $117,000 $6,000 $12,000

The awarding of items in the estate can berepresented in a matrix, as shown below.

Alan Betty Carl

House 0 1 0Boat 0 1 0Car 1 0 0



The entries in Alan’s column show which items he receives. Forexample, the 1 in Alan’s column and the car’s row indicates thatAlan receives the car. Each of the other two entries in Alan’scolumn is a 0; these zeros indicate that Alan receives neither thehouse nor the boat.

A new matrix can be computed by writing the second matrix besidethe first, as shown below.

One of the entries in the new matrix is computed by multiplyingeach entry in the first row of the first matrix by the correspondingentry in the first column of the second matrix and then finding thesum of these products.

$115,000(0) + $6,000(0) + $13,000(1) = $13,000

Because the result, $13,000, is obtained from the first row of thefirst matrix and the first column of the second matrix, it is writtenin the first row and the first column of the new matrix.

The entry in the first row and the second column of the new matrixis found by performing a similar calculation with the first row ofthe first matrix and the second column of the second matrix.

$115,000(1) + $6,000(1) + $13,000(0) = $121,000

a. Calculate the remaining entries of the new matrix.

b. The $13,000 in the first row and the first column of the newmatrix can be interpreted as the value to Alan of the items hereceived. Write an interpretation of the number in the first rowand the second column of the new matrix.

c. Write an interpretation of the number in the second row and thesecond column of the new matrix.

$ , ______ ____________ ______ ____________

13 000

_______ ______

$ , $ , $ ,$ , $ , $ ,$

115 000 6 000 13 000120 000 7 000 11 0001117 000 6 000 12 000, $ , $ ,

� � � �

0 1 00 1 01 0 00



8. Could this lesson’s estate division model encourage insincerebidding by one or more of the heirs? Explain.

9. In 1998, the U.S. Supreme Court settled a dispute between New Yorkand New Jersey over control of Ellis Island by dividing the islandbetween the two states. Is the problem of how to divide an islandamong two or more parties a continuous or discrete problem?Explain.

10. Two friends plan to share an apartment in order to obtain one thatis nicer than either could afford individually. They choose a two-bedroom apartment that rents for $1,300 monthly, includingutilities. One bedroom is larger and sunnier than the other. Proposea model for deciding which of the friends gets the nicer bedroom.

The Battle Over Ellis Island

Government TechnologyMagazine

January, 2000

Between 1892 and 1954,Ellis Island was thegateway to America forsome 17 million people ontheir way to a new life.Many Americans todayhave parents orgrandparents who arrivedthere as immigrants.

Today, Ellis Island is anational monument thatdraws 5 million visitorsannually. Althoughhistorically associatedwith New York, about 83percent of the island wasrecently declared by theU.S. Supreme Court to be

within the jurisdiction ofNew Jersey.

In 1993, New Jerseyinitiated a lawsuit againstNew York in an effort toresolve a territorialdispute over the islandthat had dragged on formore than 160 years.

Since the island is ownedby the United States, thecourt’s ruling is unlikely tohave significance foranyone but the litigants.



Computer/Calculator Exploration11. It can be instructive to examine the results of an estate division

when one or more of the bids changes. However, it is tedious toredo all the calculations several times over. Fortunately, this lesson’s estate division algorithm can be implemented on aspreadsheet, which simplifies changing values, and inspecting theresults. Use a computer spreadsheet to perform this lesson’s estatedivision algorithm. A sample output and the formulas thatgenerate it are shown below and on the next page. In this case, theresults are those of Exercise 4. Once your spreadsheet is complete,use it to answer the questions that follow.

A B C D E F

Estate Division Spreadsheet123456789

1011121314151617

Final total

Extra cashCash receivedObject valueFair shareShareBid total

TicketMotorcyclePainting

Amy Bart Carol2000.004000.00

500.00

5000.002000.00

100.00

3000.003000.00

300.00

Amy Bart Carol11500.000.333333

3833.33

12100.00

4033.33

11300.00

3766.670.333333 0.333333

0.005000.004500.00–666.67 –966.67 3766.67

955.56955.56955.56

4788.89 4988.89 4722.23

Cash:5000.00

666.67966.67

–3766.67

2866.67



a. What if the amount of cash in the estate is 0? Change theamount in cell F9 to 0 and see what happens. Describe theresult.

b. What would happen if Bart lies about the value he places on themotorcycle and says he feels it is worth $5,000? Change theamount in cell C5 to 5,000 and see. (Change the cash back to5,000 before doing this one.) Describe the result.

c. What happens if Bart really does feel that the motorcycle isworth $5,000 but accepts a $2,000 bribe from Amy to bid$2,000? How does this collusion between Bart and Amy changethe value of the final settlements for Bart and Amy?

d. Explain how to change the formulas in the spreadsheet toaccount for the situation in Exercise 5.

Projects12. Matrix calculations like the multiplication shown in Exercise 7 are

useful in programming computers to do tedious calculations.Research and report on the use of matrix applications in computerscience.

13. Research division models that are used at auctions. What areDutch and English auctions? Why do some auctions award thecontract to the second-highest bidder? When are closed and openbids used?

12345

6789

101112131415

16

Estate Division Spreadsheet

TicketMotorcyclePainting

Amy Bart Carol2000.004000.00500.00

5000.002000.00

100.00

3000.003000.00

300.00

A B C D E F

Final total

Extra cashCash received

Fair shareShareBid total

Amy Bart Carol Cash:

5000

Object value

=SUM(B4:B6)+$F$9=1/3=B9*B10=B5+B6=B11–B12

=$F$14*B10

=SUM(B12:B14)

=SUM(C4:C6)+$F$9=1/3

=C9*C10=C4=C11–C12=$F$14*C10

=SUM(C12:C14)

=SUM(D4:D6)+$F$9=1/3=D9*D10

=0=D11–D12=$F$14*D10

=SUM(D12:D14)

=–B13=–C13=–D13

=SUM(F9:F12)


Documents

Lesson 2 - Weeblygvmath.weebly.com/uploads/5/6/5/3/56539921/2.2_estate_division.pdf · Lesson 2.2 • Estate Division 65 Exercises 1. The success of a fair division model requires