1. Chapter 13: Fair Division Lesson Plan
For All Practical Purposes
Fair Division
The Adjusted Winner Procedure
The Knaster Inheritance Procedure
Fair Division and Organ Transplant Policies
Taking Turns
Divide and Choose
Cake-Division Procedures: Proportionality
Cake-Division Procedures: The Problem of Envy
Mathematical Literacy in Today’s World, 8th ed. 1
© 2009, W.H. Freeman and Company

2. Chapter 13: Fair Division Fair Division
Fair Division Problems
When demands or desires of one party are in conflict with those of another; however, objects must be divided or contents must be shared in such a way that no one is treated unfairly.
Examples include divorce, inheritance, the liquidation of a business, labor-management negotiation, or an international dispute.
Fair Division Schemes, or Procedures:
The objective in fair division problems is to devise a scheme for dividing objects (divisible or indivisible) in such a way that each of the parties (or individual players) involved obtains a share that he or she considers to be fair. 2

3. Chapter 13: Fair Division The Adjusted Winner Procedure
The Adjusted Winner Procedure – Developed in the mid-1990s, this procedure allows two parties to settle any dispute involving issues or objects by each party quantifying the importance he/she attaches to getting its own way on each of the objects or issues.
Four Basic Steps for Adjusted Winner Procedure
Each party distributes 100 points over the items in such a way that reflects their relative worth. Essentially, the party quantifies the importance for an item by placing a higher “bid” on that item.
Each item is initially given to the party that assigned it more points. Each party then assesses how many of his or her own points he or she has received. The party with the fewest points is now given each item on which both parties place the same points.
Since the point totals are not likely to be equal, let A be the party with the higher point total and B be the other party. Start transferring items from A to B in a certain order (as in step 4) until the point totals are equal.
The order is determined by going through the items in order of increasing point ratio. An item’s point ratio, fraction, is = (A’s point value) ÷ (B’s point value) , where A is the party with the higher point total. 3

4. Chapter 13: Fair Division The Adjusted Winner Procedure
Example of the Adjusted Winner Procedure
Consider the application to business mergers of two companies.
Two companies: Glaxo Wellcome (GW) and SmithKline Beecham (SKB)
Success of a merger often turns out to be what deal-makers call social issues—high power, position, sacrifice, and status are often the most difficult to allocate.
Applying the Adjusted Winner Procedure to a Merger This table shows which issues are most important:
Each item is initially given to the party that assigned it more points:
GW placed more points on headquarters (25) and chairman (35); total = 60
SKB is given name (10), CEO (35), and layoffs (25); total = 70
Next, calculate which issue (or fraction of) to transfer from SKB to GW to equalize points.
Point Allocations
Issues
Glaxo Wellcome
SmithKline Beecham
Name 5 10
Headquarters 25
Chairman 35 20
CEO 15
Layoffs
Total
100 4

7. Chapter 13: Fair Division The Adjusted Winner Procedure
Theorem: Properties of the Adjusted Winner Allocation
For two parties, the adjusted winner procedure produces an allocation, based on each player’s assignment of 100 points over the items to be divided, that has the following properties:
The allocation is equitable: Both players believe that he or she received the same fractional part of the total value.
The allocation is envy-free: Neither player would be happier with what the other received.
The allocation is Pareto-optimal: No other allocation, arrived at by any means, can make one party better off without making the other party worse off.
Pareto optimality (also named after Vilfredo Pareto) is an extremely important property to economists. The order of transfer in step 4 of the adjusted winner procedure is so important because it guarantees that the outcome is Pareto-optimal. 7

8. Chapter 13: Fair Division The Knaster Inheritance Procedure
Proposed by Bronislaw Knaster in 1945, this is an auction scheme designed for estate settlement with many heirs, assuming that each heir has a large amount of cash at his/her disposal.
Method for the Knaster Inheritance Procedure.
First, it begins by having each player (independently) assign a dollar value (a “bid”) to the item or items to be divided, so as to reflect the absolute worth of each object to that player.
The item is awarded to the person with the highest bid, and his/her “fair share” of the heir amount is subtracted from this bid. The remaining amount is put into a temporary kitty to be divided among the other heirs, according to his or her fair share.
The allocation resulting from this procedure leaves each party feeling that he or she received a dollar value at least equal to his or her fair share (and often more so).
It never requires the dividing or sharing of an object, but it may require that the players have a large amount of cash on hand. 8

9. Chapter 13: Fair Division The Knaster Inheritance Procedure
Example of the Knaster Inheritance Procedure
Suppose there is just one object in the inheritance, a house, and four heirs: Bob, Carol, Ted, and Alice.
Begin with each heir bidding (simultaneously and independently) on the house. The bids are:
Bob
Carol
Ted
Alice
$120,000
$200,000
$140,000
$180,000
Since Carol has the highest bid, the house is awarded to her.
Her fair share is only ¼ of the $200,000 she thinks the house is worth. Then she places ¾ of her bid of $200,000 (amount of $150,000) into the temporary “kitty.”
Each of the other heirs withdraws his/her fair share (¼ of their bid) from the kitty. The amount left over is divided among all.
Bob withdraws $120,000/4 = $30,000
Ted withdraws $140,000/4 = $35,000
Alice withdraws $180,000/4 = $45,000
Total withdrawn is $110,000
Surplus in the kitty is: $150,000 − 110,000 = $40,000
This amount of $40,000 is divided among the four heirs ($10,000 extra to each). 9

14. Chapter 13: Fair Division Fair Division and Organ Transplant Policies
U.S. Congress passed National Organ Transplant Act in 1984 to increase equity in national system of organ allocation. This was based on three criteria:
1. Waiting time. List is made according to how long each person has been waiting for a transplant. Each person receives 10 times the calculated fraction of people on the list at or below his/her position in the list.
2. Suitability. Each donor and potential recipient have 6 relevant antigens matched and receive 2 points for each successful match.
3. Disadvantage. Each person has antibodies that rule out a certain percentage of the population as being potential donors. Each recipient is awarded 1 point for each 10% of the population they are “sensitized” against.
Peyton Young pointed out the “priority paradox” in the case where the next person on the list may not receive the next organ once the first person is removed and the point system recalculated.

15. Chapter 13: Fair Division Taking Turns
With two parties, one party selects an object, then the other party selects one, then the first party again, and so on.
Questions arise with taking turns:
How do we decide who goes first?
Possible answers: Toss a coin or bid for the right to go first.
Since choosing first is often quite an advantage, should the other party be compensated in some way?
Possible compensation could be extra choices at the next turn.
Should a player always choose his/her favorite from those that remain, or try to choose strategically to get better overall results?
If a player chooses strategically, he/she could get better results.
Taking Turns – A fair-division procedure in which two or more parties alternate selecting objects. 15

16. Chapter 13: Fair Division Taking Turns
Example of Taking Turns
Suppose Bob and Carol are getting a divorce and their four main possessions are ranked from best to worst by each:
If they both choose sincerely, the items would be allocated as follows:
First turn: Bob takes the pension. Second turn: Carol takes the house. Third turn: Bob takes the investments. Fourth turn: Carol is left with vehicles.
Bob’s Ranking
Carol’s Ranking
Best
Pension
House
Second best
Investments
Third best
Worst
Vehicles
Suppose Bob votes insincerely and chooses the house first. He actually would do better (getting his first, second and third choice).
First turn: Bob takes the house Second turn: Carol takes the investments Third turn: Bob takes the pension Fourth turn: Carol is left with vehicles.
Bottom-up strategy is another method that can be used for optimal strategy for rational players, each knowing the other’s preferences. 16

19. Chapter 13: Fair Division Divide and Choose
The origins of the divide and choose procedure date back thousands of years.
The rules to the procedure are simple: Someone divides the object into two parts and the other person chooses first.
Divide and Choose – A fair division procedure for dividing an object into two parts in any way that one desires and the other party chooses which-ever part he/she wants.
The natural strategies of the divide and choose procedure are quite obvious:
The divider makes the two parts equal in his estimation.
The chooser selects whichever piece he feels is more valuable.
Other strategic considerations may be relevant, such as the question Would you rather be the divider or the chooser? Not knowing the other’s preference, one would want to be the chooser. 19

20. Chapter 13: Fair Division Cake-Division Procedure: Proportionality
This procedure is used to divide and choose some object among three or more people, and cake is the metaphor used to help explain this procedure.
The cake-division procedure involves finding allocations of a single object that is finely divisible. In this procedure, each player has a strategy that will guarantee his/her “satisfaction,” even in the face of collusion by the others.
To be “satisfied” means that we “think our piece is of size or value at least what we viewed as a fair share” or “we do not want to trade.” 20

21. Chapter 13: Fair Division Cake-Division Procedure: Proportionality
Definition of Proportional and Envy-Free
A cake-division procedure (for n players) will be called proportional if each player’s strategy guarantees that player a piece the size or value at least 1/n of the whole in his or her own estimation.
It will be called envy-free if, as in the adjusted winner context, each player’s strategy guarantees that player a piece he or she considers to be at least tied for largest or most valuable. 21

22. Chapter 13: Fair Division Cake-Division Procedure: Proportionality
Lone-Divider Method – (Hugo Steinhaus)
A cake-division procedure that works for three players and produces an allocation that is proportional but not, in general, envy-free.
One cuts the cake and the other two players must “approve a piece” if they think it is the size of at least one-third.
If the other two players “disapprove of a piece,” then that piece can go to the cutter, or some pieces can be put back together and recut.
Last-Diminisher Method – (Banach and Knaster)
A cake division procedure for any number of players that produces an allocation that is proportional but not, in general, envy-free.
For four or more players, one person cuts a piece of the cake and hands it to another player.
The player examines it. If it looks fair, he passes it along to the next player. If it looks too big, he trims it first (puts the trimmed part back on the cake) and then passes it on (either unaltered or diminished).
This continues down the line until all everyone has had a chance to trim.
The last person to trim the piece receives that piece and exits the game. 22

28. Chapter 13: Fair Division Cake-Division Procedure: The Problem of Envy
In the lone-divider and last-diminisher methods, envy could easily exist among the players.
Example: The person who initially cuts the cake feels he or she is making the cut even. However, as the pieces are either put back together and re-cut, or trimmed and passed on, the original portions change, and so they may be perceived differently by the original cutter—envy may occur.
Selfridge-Conway Envy-Free Procedure (Three Players)
This procedure involves constructively obtaining three pieces of cake through successive trimming by the players (and the trimmings can be allocated later).
At the conclusion, each of the three people thinks he or she received a piece at least tied for the largest with this method.
This procedure is envy-free, as well as proportional. 28