Chapter 13: Fair Division
Lesson Plan








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
© 2009, W.H. Freeman and Company
For All Practical
Purposes
Mathematical Literacy in
Today’s World, 8th ed.
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.
Chapter 13: Fair Division
The Adjusted Winner Procedure
 The Adjusted Winner Procedure – Developed in the mid1990s, 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
1.
2.
3.
4.
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.
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:
Point Allocations
Issues
Glaxo Wellcome SmithKline Beecham
Name
5
10
Headquarters
25
10
Chairman
35
20
CEO
15
35
Layoffs
20
25
Total
100
100
 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.
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.
Chapter 13: Fair Division
The Knaster Inheritance Procedure
 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.
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).
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.
Chapter 13: Fair Division
Taking Turns
 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.
Taking Turns – A fairdivision procedure in
which two or more
parties alternate
selecting objects.

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.
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
Bob’s Ranking Carol’s Ranking
items would be allocated as follows:
Best
Pension
House
Second best
House
Investments
Third best
Investments
Pension
Worst
Vehicles
Vehicles
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.
 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.
Chapter 13: Fair Division
Divide and Choose
 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.
Chapter 13: Fair Division
Cake-Division Procedure: Proportionality
 Cake-Division Procedure
 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.”
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.
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.
Chapter 13: Fair Division
Cake-Division Procedure: The Problem of Envy
 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.