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:

Glaxo Wellcome/SmithKline Beecham Merger IssueGWSKB Name510 Headquarters2510 Chairman3520 CEO1535 Layoffs2025 Total100

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.

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. 1 st 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 simultaneously- submit 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 BobCarolTedAlice 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 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 Inheritance BobCarolTedAlice $30,000House-$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: BobCarolTedAlice $40,000House-$140,000$45,000$55,000

Now the cabin… BobCarolTedAlice 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 Ted receives the Cabin and places $67,500 in the kitty. BobCarolTedAlice $15,000$10,000Cabin-$67,500$12,500 The $30,000 surplus is split evenly 4 ways, so each person gets an additional $7,500

Cabin & Boat BobCarolTedAlice $22,500$17,500Cabin-$60,000$20,000 Practice by trying the same for the boat: BobCarolTedAlice Boat-$20,875$7,625$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 A5210 B C400 D2360 E1690 Potential Recipient Months Waiting Antigens Matched Percent Sensitized Totals A B84214 C6006 D46616 E Points

Taking Turns Mostly common sense But… Who gets to choose first? How do we compensate the 2 nd chooser for have the disadvantage? Are there an special strategic considerations to take into account?

Bottom-Up Strategy Bob’s rankingCarol’s Ranking BestPensionHouse 2 nd BestHouseInvestments 3 rd BestInvestmentsPension WorstVehicles 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 rankingCarol’s Ranking BestPensionHouse 2 nd BestHouseInvestments 3 rd BestInvestmentsPension WorstVehicles Bob Carol VehiclesInvestments PensionHouse

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 1/4 th of the cake. If Bob thought it was more than 1/4 th, 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 1/4 th. And so on…

Selfridge-Conway Envy- 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