Fair DivisionCh. 13 Finite Math

Fair divisionFair division There are about 1.2 million

divorces every year in the U.S. alone.

International disputes redefine borders between nations.

No one likes to be treated unfairly, so we search for a mathematical way to keep things fair.

Adjusted Winning Adjusted Winning ProcedureProcedure

Developed in the mid-1990s, this procedure lets two parties settle any dispute with

certain mathematical guarantees of “fairness.”

Adjusted Winning Adjusted Winning Procedure (basic Procedure (basic

steps)steps)

1) Each party distributes 100pts over the items in a way that reflects their relative

worth to the party

2) Initially, each item is assigned to the party that assigned it more points. Each party then assess how many of his or her own points he or she has received. The party with the fewest points is now given items on which both parties placed the same amount of points.

Adjusted Winning Adjusted Winning Procedure (basic Procedure (basic

steps)steps)

3) Since the point totals are not likely to be equal, let A denote the party with the higher

total and B be the other part. Start transferring items from A to B, in a certain order, until the point totals are equal. The

last item transferred may be a fraction of an item.4) The order in which this is done is

extremely important and is determined by going through the items in order of increasing point ratio:

€

point ratio =A's point value of item

B's point value of item

Glaxo Wellcome/SmithKline Glaxo Wellcome/SmithKline Beecham MergerBeecham Merger

Issue GW SKB

Name 5 10

Headquarters 25 10

Chairman 35 20

CEO 15 35

Layoffs 20 25

Total 100 100

Splitting an itemSplitting an item Layoffs are the first to be split by the companies because

of their low point ratio.

Giving the whole issue would just make it unfair for the other company, so it must be broken into a fraction.

€

10 + 35 + 25x = 25 + 35 + 20 1− x)( )

€

45 + 25x = 60 + 20 − 20x

45 + 25x = 80 − 20x

45x = 35

x =35

45=

7

9

EquitableEquitableA fair-division procedure, like adjusted winner, is said to be

equitable if each player believes he or she received the same

fractional part of the total value.

Envy-FreeEnvy-FreeA fair-division procedure is said to be envy-free if each player has a strategy that can guarantee him

or her a share of whatever is being divided that is, in the eyes of that player, at least as large as that received by any other player, no matter what the other players

do.

Pareto-OptimalPareto-OptimalA fair-division procedure is said to be Pareto-Optimal if it produces an allocation with the property

that no other allocation, achieved by any means whatsoever, can make any one player better off

without making some other player worse off.

The Knaster The Knaster Inheritance ProcedureInheritance Procedure

Adjusted winning procedure is great for 2 heirs

The Knaster Inheritance Procedure can be used with more than two heirs. 1st proposed by Bronislaw Knaster in

1945

Major drawback: It requires the heirs to have a large amount of cash at their disposal

The Knaster The Knaster Inheritance ProcedureInheritance Procedure

For each object, the following steps are performed:

1) The heirs-independently and simultaneously- submit monetary bids for

the object2) The high bidder is awarded the object, and he or she places all but 1/n of his or her

bid in a kitty.

So, if there are 4 heirs (n=4), then he or she places all but one-fourth– that is, 3/4ths– of his or her bid in the kitty

The Knaster The Knaster Inheritance ProcedureInheritance Procedure

3) Each of the other heirs withdraws from the kitty 1/n of his or her bid.

4) The money remaining in the kitty is divided equally among the n heirs.

A Four-Person A Four-Person InheritanceInheritance

Bob Carol Ted Alice

House $120,000

$200,000

$140,000

$180,000

Cabin $60,000 $40,000 $90,000 $50,000

Boat $30,000 $24,000 $20,000 $20,000

Initial Bids

Carol gets the house. Since n=4, Carol must pay all but 1/n of her bid to a kitty. The other 3 bidders withdraw 1/n of their

bids from this amount.

4-Person Inheritance4-Person Inheritance

Bob Carol Ted Alice

$30,000 House-$150,000

$35,000 $45,000

Carol places $150,000 in the kitty (all but one-fourth of her original bid).

This leaves $40,000 remaining after the withdraws. This total is split evenly between all

bidders. Each walks away with the following:

Bob Carol Ted Alice

$40,000 House-$140,000

$45,000 $55,000

Now the cabin…Now the cabin…Bob Carol Ted Alice

House $120,000

$200,000

$140,000

$180,000

Cabin $60,000 $40,000 $90,000 $50,000

Boat $30,000 $24,000 $20,000 $20,000

Ted receives the Cabin and places $67,500 in the kitty.

Bob Carol Ted Alice

$15,000 $10,000 Cabin-$67,500 $12,500

The $30,000 surplus is split evenly 4 ways, so each person gets an additional $7,500

Cabin & BoatCabin & Boat

Bob Carol Ted Alice

$22,500 $17,500 Cabin-$60,000 $20,000

Practice by trying the same for the boat:

Bob Carol Ted Alice

Boat-$20,875 $7,625 $6,625 $6,625

Bob: Boat+$41,625Carol: House-$114,875

Ted: Cabin-$8,375Alice: $81,625

Taking Turns: Taking Turns: Transplant Waiting ListTransplant Waiting List

For the first 15 minutes of class, come up with a fair way to decide who gets the first available organ when many people all

across the country may need it to survive.

Fair Division and Fair Division and Transplant PoliciesTransplant Policies

In 1984, The U.S. Congress passed the National Organ Transplant Act.

First come, first serve?

Whoever needs it the most?

Should you get it if you are more compatible with the organ?

Organ Procurement Organ Procurement and Transplantation and Transplantation

NetworkNetworkCriterion 1) Waiting time: for each recipient, one calculates the fraction of people at or below their

waiting time. The recipient gets 10 times that fraction of points

Criterion 2) Suitability: The donor and recipient each have 6 relevant antigens that are ether

matched or not matched, with the likelihood of a successful transplant increasing with more

matches. Two points are awarded for each match.Criterion 3) Disadvantage: Each person has

antibodies that may make them unable to receive a certain donor’s organ. For each 10% of the

population that a recipient is “sensitized against,” they get 1 point.

OPTNOPTNPotential Recipient

Months Waiting

Antigens Matched

Percent Sensitized

A 5 2 10

B 4.5 2 20

C 4 0 0

D 2 3 60

E 1 6 90Potential Recipient

Months Waiting

Antigens Matched

Percent Sensitized

Totals

A 10 4 1 15

B 8 4 2 14

C 6 0 0 6

D 4 6 6 16

E 2 12 9 23

Points

Taking TurnsTaking Turns Mostly common sense

But… Who gets to choose first? How do we compensate the

2nd chooser for have the disadvantage?

Are there an special strategic considerations to take into account?

Bottom-Up StrategyBottom-Up StrategyBob’s ranking Carol’s

Ranking

Best Pension House

2nd Best House Investments

3rd Best Investments Pension

Worst Vehicles Vehicles

Say that Bob is going to pick first. He knows that Carol’s least favorite is the Vehicles, so he would only pick that for his last choice even if he really wants it. He does not have to worry

because Carol doesn’t want it.

Bottom-Up StrategyBottom-Up StrategyBob’s ranking Carol’s

Ranking

Best Pension House

2nd Best House Investments

3rd Best Investments Pension

Worst Vehicles Vehicles

Bob

Carol

1

2

3

4

VehiclesInvestments

PensionHouse

Divide & ChooseDivide & Choose

Would you rather be the divider or

the chooser?

Cutting the Cake Cutting the Cake (a metaphor)(a metaphor)

Cake-Division: Cake-Division: Proportional ProcedureProportional Procedure

Bob, Carol, and Ted will get pieces X, Y, and Z of cake. If Bob cuts the cake, Carol

“approves” of piece Y, and Ted “approves” of piece Z, then there is no problem.

Cake-Division:Cake-Division: Lone-divider MethodLone-divider Method

If Carol and Ted only approve of piece X, then X and Y are rejoined for Carol and Ted to

divide and choose while Bob gets piece Z.

Last –Diminisher Last –Diminisher MethodMethod

Carol, Bob, Ted, and Carol pass around the piece of cake that Carol cut and assumed to

be 1/4th of the cake. If Bob thought it was more than 1/4th, he trims some and puts the

trimmings back on the cake. The cake is passed to everyone. The last person to trim it eats it because all will have greed that it is at

least 1/4th. And so on…

Selfridge-Conway Selfridge-Conway Envy-Free ProcedureEnvy-Free Procedure

Player 1 cuts the cake into 3 piece that he or she considers to be the same size. He or she then hands the pieces to player 2

Player 2 trims at most one of the three pieces to create at least a two-way tie for largest. Setting the trimmings aside, player two hands the three pieces to player

3.

Player 3 chooses one piece that he or she feels to be at least tied for largest

Player 2 chooses from the remaining pieces. If the piece she trimmed remains, she must take it.

Player 1 gets the remaining piece

Let player 2 cut the trimmings into 3 “equal” pieces. Then, let the players choose in the

following order: 3,1,2