Chapter 4. Section 4.1: Divide and Choose Methods. Next: reading homework

Transcription

1 Chapter 4 Section 4.1: Divide and Choose Methods Next: reading homework

2 Reading Homework Read Section 4.2 Do problem 22 Next: fair division

3 Fair Division Mathematical way of discussing how to divide resources Ideally, the parties doing the dividing will be satisfied with the results Next: some terminology

4 Terminology People who are trying to share the resource(s) are called players. A solution of a fair-division problem is called a fair-division procedure or fair-division scheme. Next: different types of fair division problems

5 Types of fair-division problems Continuous fair-division problem: can split the object(s) to be divided into pieces of any size with no loss of value E.g.: Cutting a cake Discrete fair-division problem: objects to be shared cannot be subdivided Assume the players do not want to just sell the objects and divide the proceeds E.g.: Joint owners of a car, a house, and a boat seeking a fair way to split the assets and go their separate ways Next: one more type of fair division problem

6 Types of fair-division problems Mixed fair-division problem: may subdivide some objects to be shared but not others. Basically a combination of continuous and discrete fair division problems E.g.: Heirs with an estate that contains money as well as a house, car, etc., that they do not wish to sell Next: some assumptions

7 Assumptions The value of a player s share is determined by his or her own preferences or values. E.g., cutting a cake -- big/small piece, more/less frosting, etc. Next: terminology; what is a fair share?

8 When is a share fair? In a fair division problem with n players, a player has received a fair share if that player considers his/her share to be worth at least 1/n of the total value being shared. A division that results in every player receiving a fair share is called proportional. Next: back to assumptions

9 Assumptions The value of a player s share is determined by his or her own preferences or values. A player s values in a fair-division problem cannot change based on the results of the division. No player has knowledge of any other player s values. E.g., cutting a cake -- big/small piece, more/less frosting, etc. Next: fair division w/ 2 players

10 Two players Two players, X and Y, divide a cake. Player X divides the cake into two pieces that he or she considers of equal value (X is the divider). Player Y picks the piece he/she considers of greater value (Y is the chooser). Player X gets the piece Y did not select. In player Y s eyes, one of the pieces has at least half the value, so this is a fair way to do it. Next: a more interesting example w/ 2 players

11 An example Suppose Jerry and George are sharing a $4 pizza which is half pepperoni and half cheese. Jerry is easygoing and likes both equally; George likes cheese 4 times as much as he likes pepperoni. Jerry cuts the pizza into 6 equal slices and arranges 3 on one plate (2 cheese and 1 pepperoni) and 3 on another. Which plate does George choose? Ask class: who is divider/chooser? Also, how would original halves be valued? Work out on board: the two pieces the pizza has been divided into are the two plates. Each plate is worth $2 to Jerry George would consider the cheese half to have 80% ($3.20, $1.067/piece) of the value and the pepperoni half to have 20% ($0.80, $0.267/piece). For George, the plate with 2 cheese has value $2.40 and other has value $1.60. Next: divide-and-choose for 3 players

12 Today Finish Section 4.1: Divide and Choose Methods Start Section 4.2: Discrete and Mixed Division Problems Next: reading homework

13 Writing Assignment 5 Due Wednesday, November 11 Stress the difference between discrete and continuous objects Next:

14 Divide-and-Choose for Three players Three players, X, Y, and Z, divide a cake. 1. Player X divides the cake into three pieces of equal value to him/her. 2. Players Y and Z decide independently which pieces are worth at least 1/3 the cake s value. 3. Players Y and Z announce which pieces are acceptable. 4. Depends on which pieces are declared acceptable: a. If at least 1 piece is unacceptable to either Y or Z, then X gets one of those unacceptable pieces. If Y and Z can choose different acceptable pieces from the 2 remaining, they do so. If not, they put them back together and use the 2-player method. b. If every piece is acceptable to Y and Z, then both Y and Z take acceptable pieces and X gets whatever is left. Notice, in Step 2, that for each chooser, not all of the pieces can have less than 1/3 the value; there must be at least one piece that Y finds acceptable, and so for Z. Divide-and-choose can be extended to more players, but it gets more complicated. Next: an example with 3 players

15 An example Suppose Courtney, Nathan, and Ben want to fairly divide 24 ounces of Neapolitan ice cream made up of equal amounts of vanilla, chocolate, and strawberry. Suppose... Courtney likes each flavor equally well Nathan likes chocolate by a ratio of 2-to-1 over both strawberry and vanilla (i.e., a 1 to 2 to 1 ratio) Ben likes vanilla to chocolate to strawberry in a ratio of 1 to 2 to 3. If Courtney is the divider, what is the result of the divide-andchoose method for 3 players? Assume Courtney divides into the 3 8-oz. chunks of single flavors. Notice: a fair share for Courtney is 8 pts, a fair share for Nathan is 10 2/3, and for Ben it s 16 pts, assuming 1 pt/oz. Next: courtney divides, and we use points to keep track of things

16 Step 1 Divider divides the ice cream: Courtney does not prefer a flavor, so suppose she divides the ice cream into 3 equal parts, each consisting of one of the flavors. Assign points: 1 pt/oz, for a total of 24 points. Next: Nathan s value system

17 Step 2 Choosers determine acceptable pieces: Nathan s preference ratio, 1 to 2 to 1, means that he might assign 1 pt/oz of vanilla, 2 pts/oz of chocolate, and 1 pt/oz of strawberry for totals of 8, 16, and 8. This is 32 points total. What is a fair share in Nathan s value system? Must be worth at least 10 ⅔ points. Next: ben s value system

18 Step 2 Choosers determine acceptable pieces: Ben s preference ratio, 1 to 2 to 3, means that he might assign 1 pt/oz of vanilla, 2 pts/oz of chocolate, and 3 pt/oz of strawberry for totals of 8, 16, and 24. This is 48 points total. What is a fair share in Ben s value system? Must be worth at least 16 points. Next: steps 3 and 4

19 Steps 3 and 4 Step 3: choosers reveal which pieces are acceptable Step 4: Portion 1 is unacceptable to both choosers, so Courtney gets Portion 1. Only Portion 2 is acceptable to Nathan, so he gets it, and Ben gets Portion 3. Next: another way, the last diminisher method

20 Last-Diminisher Method Suppose some number of players X, Y, are dividing a cake. 1. Player X cuts a piece of the cake that he/she deems fair. 2. Each player judges the fairness of the piece. A. If a player considers the piece to be fair or less than fair, then it is the next player s turn to judge. B. If a player considers the piece to be larger than fair, then that player trims the piece to a smaller size that he/she deems fair. The trimmedoff piece is reattached to the main body of the cake, and it is the next player s turn to to judge the new piece. 3. The last player who trimmed the cake to a smaller size gets the piece. If no player trimmed it, then player X gets the piece. 4. Repeat until only two are left, then use the divide-and-choose method. Motivation for a player to trim a piece if her/she considers it to be more than a fair share is that if it is not trimmed, someone else will get the excessively large piece. Assume, though, that trimmers always try to create a fair share. Next: revisit neapolitan example

21 I scream redux Suppose Courtney, Nathan, and Ben look to divide another 24 oz. of Neapolitan ice cream. Suppose Ben (with preference ratio of 1 to 2 to 3) is the divider. He places a total value of 48 points on the system. Thus a fair share is worth 16 points to him, e.g., 8 oz. of vanilla and 4 oz. of chocolate Next: judging

22 I scream redux Next, the others judge fairness of 8 oz. of vanilla, 4 oz. of chocolate. Suppose Courtney judges first. She places a total value of 24 points on the system, so a fair share (to her) is 8 points. 16 points is too much, so she removes 4 points worth, say the 4 oz. of chocolate Next: nathan finishes juding

23 I scream redux Nathan finishes the judging. He places a value of 32 points on the system. A fair share has 10 ⅔ points 8 oz. of vanilla is worth 8 points, so Nathan does not trim it. Next: courtney gets the 8 oz., and end

24 I scream redux Last diminisher gets the piece: Courtney gets the 8 oz. of vanilla Nathan and Ben use the 2-player divide-andchoose method to divide the remaining ice cream Point out that there are different ways of doing it at each of these points Next: end

25 Chapter 4 Section 4.2: Discrete and Mixed Division Problems Next: reading homework

26 Question How should people divide/share discrete (not easily subdivided) objects? Examples Car House Boat Stress the difference between discrete and continuous objects Next:

27 Method of Sealed Bids Any number of players, n, are to share any number of items, with the possibility of monetary compensation to ensure fairness. 1. All players independently submit bids, each stating a monetary value for each item. 2. Each item goes to the highest bidder, and that player contributes the dollar amount of his/her bid to a compensation fund. 3. From the compensation fund, each player receives 1/n of his/her bid on each item. 4. Any money left in the compensation fund is distributed equally to all players. Notice that it is in everyone s best interest to bid fairly since no player knows if he/she is buying or selling each item.

28 Next time... Example

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