fair division ch. 13 finite math. fair division there are about 1.2 million divorces every year in...
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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