Fair Division
Ch. 13 Finite Math
Fair 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
Procedure
Developed in the mid-1990s, this procedure lets two
parties settle any dispute with certain mathematical
guarantees of “fairness.”
Adjusted Winning
Procedure (basic 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
Procedure (basic 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:
A's point value of item
point ratio 
B's point value of item
Glaxo Wellcome/SmithKline
Beecham Merger
Issue
GW
SKB
Name
5
10
Headquarters
25
10
Chairman
35
20
CEO
15
35
Layoffs
20
25
Total
100
100
Splitting 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 7

45 9
Equitable
A 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-Free
A 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-Optimal
A 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 Inheritance
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 Inheritance
Procedure
For each object, the following steps are performed:
1) The heirs-independently and simultaneouslysubmit monetary bids for the object
2)
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 Inheritance
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 Inheritance
Initial
Bids
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
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 Inheritance
Carol places $150,000 in the kitty (all but one-fourth of her
original bid).
Bob
Carol
Ted
$30,000
House-$150,000 $35,000
Alice
$45,000
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
$40,000
House-$140,000 $45,000
Alice
$55,000
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 & 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+$41,625
Carol: House-$114,875
Ted: Cabin-$8,375
Alice: $81,625
Taking Turns: Transplant
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
Transplant 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 and
Transplantation Network
Criterion 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.
OPTN
Potential
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
90
Points
Potential
Recipient
Months
Waiting
Antigens
Matched
Percent
Totals
Sensitized
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
Taking 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 Strategy
Bob’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 Strategy
Bob’s ranking
Carol’s Ranking
Best
Pension
House
2nd Best
House
Investments
3rd Best
Investments
Pension
Worst
Vehicles
Vehicles
Bob
House
Pension
1
Carol
3
Investments
2
Vehicles
4
Divide & Choose
One party divides the object
into two parts in any way
that he or she desires, and
the other party chooses
whichever part he or she
wants.
Would you rather be
the divider or the
chooser?
Cutting the Cake
(a metaphor)
Simple, but what if I want
the piece with the big glob
of icing on it and not too
much chocolate?
Cake-Division:
Proportional Procedure
A cake-division procedure is considered proportional
(for all n players), if each player’s strategy guarantees
that player a piece of size or value at least 1/n of the
whole in his or her own estimation.
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:
Lone-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 Method
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 EnvyFree Procedure

Player 1 cuts the cake into 3 piece that he or she considers to be the same
size.


Player 2 trims at most one of the three pieces to create at least a two-way
tie for largest.


He or she then hands the pieces to player 2
Setting the trimmings aside, player two hands the three pieces to player 3.
Let player 2 cut the trimmings into 3 “equal” pieces.
Player
piece that
he or in
shethe
feelsfollowing
to be at leastorder:
tied for 3,1,2
largest
Then,3 chooses
let the one
players
choose

Player 2 chooses from the remaining pieces. If the piece she trimmed
remains, she must take it.

Player 1 gets the remaining piece