Chapter 13: Fair Division

Chapter 13: Fair Division

October 9, 2013


Last time

Adjusted Winner Procedure

Knaster’s Inheritance procedure



Basic Steps in the Adjusted Winner Procedure

Step 0: Each party distributes 100 points over the items in a way thatreflects each item’s relative worth to that party.

Step 1: Each item on which the assigned points differ is initially givento the party that assigned it more points. Add up the totalnumber of points each party feels that he or she has received.The party with the fewest points is now given all the items onwhich both parties placed the same number of points. Onceagain, add up the total number of points each party feels thathe or she has received. The party with the most points iscalled the initial winner; the other party is called the initialloser.

Step 2: For each item given to the initial winner, calculate the “pointratio”



Basic Steps in the Adjusted Winner Procedure Cont.

Step 3: Start moving items from the initial winner to the initial loserin ascending order of point ratio. Stop when you get to anitem whose move will cause the initial winner to have fewerpoints than the initial loser. This item will need to be split orshared and is thus called the shared item.

Step 4: Let x represent the fractional part of the shared item that willbe moved from the initial winner to the initial loser. Write aformula that equates each party’s total points after thesharing of this item.

Step 5: Solve the equation and state the final division of itemsbetween the two parties.


Problem

Suppose that Calvin and Hobbes discover a sunken pirate ship andmust divide their loot. How should they divide their loot using theadjusted winner procedure.

Object Calvin’s Points Hobbes’s PointsCannon 10 5Anchor 10 20Unopened Chest 15 20Doubloon 11 14Figurehead 20 30Sword 15 6Cannon ball 5 1Wooden leg 2 1Flag 10 2Crow’s nest 2 1


Answer

Calvin is initially awarded the Cannon(10), Sword(15), CannonBall(5), Wooden Leg(2), Flag(10), Crow’s nest(2)Hobbes is initially awarded the Anchor(20), Unopened Chest(20),Doubloon(14), Figurehead(30)Calvin’s points, 10 + 15 + 5 + 2 + 10 + 2 = 44 Initial LoserHobbes’s points, 20 + 20 + 14 + 30 = 84 Initial Winner


Answer

Point ratioAnchor 20/10 = 2Unopened Chest 20/15 = 1.333Doubloon 14/11 = 1.2727Figurehead 30/20 = 1.5The Doubloon is then transfer to Calvin and the points are now:Calvin’s points, 55Hobbes’s points, 70The Unopened Chest is transferred next, however if it is transferredthe points will be.Calvin’s points, 70Hobbes’s points, 50Which will switch the initial winner and initial loser, so the itemmust be shared.


Answer

So we must setup and equation to split the item

55 + 15x = 70 − 20x

35x = 15

x = 3/7

x = 0.428571429


Knaster’s Inheritance procedure

Basic Steps in Knaster’s Inheritance Procedure with n Heirs

For each object, the following steps are performed:

Step 1: The heirs- independently and simultaneously- submitmonetary bids for the object.

Step 2: The high bidder is awarded the object, and he or she places allbut 1/n of his or her bid in a kitty. So, if there are four heirs(n=4), then he or she places all but one-fourth- that is,three-fourths-of his or here bid in a kitty.

Step 3: Each of the other heirs withdraws from the kitty 1/n of his orher bid.

Step 4: The money remaining in the kitty is divided equally amongthe n heirs.


Problem

John and Mary inherit their parent’s old house and classic car.John bids $28,225 on the car and $55,900 on the house. Mary bids$32,100 on the car and $59,100 on the house. How should theyarrive at a fair division?


Problem Answer

Mary get the car and the house. She places 321002 + 59100

2 in thekitty. Then John withdraws 28225

2 + 559002 from the kitty. They

split the remainder. The net effect is Mary gets the car and thehouse and pays John $43,831.25.


This Time

Fair division and Organ Transplant Policies

Taking turns


Motivating Problem

What is a fair way to allocate organs through the OrganProcurement and Transplantation Network (OPTN)?

What are some things that need to be taken into account?

Waiting Time

Suitability

Disadvantage


Considerations for Organ Transplant

Waiting TimeFor each potential recipient, one calculates the fraction ofpeople at or below the spot on the list he or she occupies andthen awards that person a number of points equal to 10 timesthat fraction. The first person gets ten points the nth personout of P people gets 10(p − n + 1)/p points

SuitabilityThe donor and potential recipient each have six relevantantigens that are either matched or not matched. Two pointsare awarded for each match.

DisadvantageEach person has antibodies that rule out a certain percentageof the population as being potential donors for that person.Potential recipients are awarded 1 point for each 10 percent ofthe population they are “sensitized against”.


Example

Let’s assume we have five potential recipients, A,B,C ,D, and E ,with the following characteristics:

Potential Months Antigens PercentRecipient Waiting Matched Sensitized

A 5 2 10B 4.5 2 20C 4 2 0D 2 3 60E 1 6 90


Example cont.


A 5 2 10B 4.5 2 20C 4 2 0D 2 3 60E 1 6 90

Potential Months Antigens Percent TotalRecipient Waiting Matched Sensitized Points

A 10 4 1 15B 8 4 2 14C 6 4 0 6D 4 6 6 16E 2 12 9 23

So E gets the kidney.


Example cont.

What happens if there is a second kidney a little later?Potential Months Antigens Percent TotalRecipient Waiting Matched Sensitized Points

A 10 4 1 15B 7.5 4 2 13.5C 5 4 0 5D 2.5 6 6 14.5

So A would get the second kidney, not D.This is an example of “priority paradox”


Motivating Question

How should a group of friends pick teams to play a game ofultimate Frisbee?


Taking turns

Calvin’s Ranking Hobbes’s RankingBest Cannon SwordSecond best Sword Unopened ChestThird best Unopened Chest CannonWorst Anchor Anchor

First turn: Calvin takes the CannonSecond turn: Hobbes takes the SwordThird turn: Calvin takes the Unopened ChestWorst turn: Hobbes takes the Anchor


Taking turns

That’s not the only way Calvin and Hobbes can choose.Calvin’s Ranking Hobbes’s Ranking

Best Cannon SwordSecond best Sword Unopened ChestThird best Unopened Chest CannonWorst Anchor Anchor

First turn: Calvin takes the SwordSecond turn: Hobbes takes the Unopened ChestThird turn: Calvin takes the CannonWorst turn: Hobbes takes the AnchorSo Calvin did better by not choosing his first choice.


Bottom-up Strategy

Calvin HobbesA CB EC DD AE B

Bottom-up Strategy:Calvin: C A B

Hobbes: D E


Motivating Question

How should we divide up a piece of cake between two people?


Divide-and-Choose

Divide-and-Choose

With divide-and-choose, one party divides the object into twoparts in any way that he desires, and the other party chooseswhichever part wants.

Question: Would you rather be the divider or the chooser?


Problem

Which of the four recipients should get the kidney with thefollowing characteristics:


A 9 2 20B 6 3 0C 5 4 40D 2 6 60


Problem

Mark and Fred have inherited a number of items from theirparent’s estate, with no indication of who gets what. They rankthe items from most preferred to least preferred as follows:

Mark Fred

Truck BoatTractor Tractor

Boat CarCar Truck

Tools MotorcycleMotorcycle Tools

Assume that Mark and Fred use the bottom-up strategy and thatMark gets to choose first. Determine Mark’s first choice and thefinal allocation.


Next time

Next time we will talk about

Cake-Division

Quiz next time will be over 13.1 -13.5Patrice problems Pages 476-480, exercises 1-5, 10-14, 15, 18-23, 26


Chapter 13: Fair Division

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