Unit 6 Notes Day 6 FAIR DIVISION ALGORITHMS CONTINUOUS CASE SECTION 2.5

Transcription

1 Unit 6 Notes Day 6 FAIR DIVISION ALGORITHMS CONTINUOUS CASE SECTION 2.5

2 Warm-Up Get out: Notebook Paper for Test 5 Corrections Put phones in pockets!! Last night s HW opened up Packet p. 9

3 Warm-Up = Test Unit 5 Corrections On Notebook Paper Title the page For each Question missed, retry it show your work! Ask Questions! Remember to put your name on it!! Staple Corrections ON TOP of your test! Put phones in pockets!!

4 Questions from HW?

5 Tonight s HW Complete Derivatives Sheet Read Power Point & Finish Today s Notes handout

6 Notes Fair Division Continuous Case S E C T I O N 2. 5

7 FAIR DIVISION There are many circumstances in which the division of an object or a set of assets in a fair way is important. Examples of things that need to be fairly divided: 1. Food among children 2. A house in an estate among heirs 3. Seats in a governmental body among districts

8 FAIR DIVISION Consider these three scenarios. 1. Martha and Roy want to divide the last piece of the cake that their mother baked yesterday. Propose at least three methods of dividing the piece of cake that will seem fair to both Martha and Roy. 2. Juan and Mary are the only heirs to their mother s estate. The only object of significant value is the house in which they were raised. Propose at least three methods of resolving the issue of the disposition of the house that will seem fair to both Juan and Mary. 3. The sophomore, junior, and senior classes at Central High School have 333, 288, and 279 members, respectively. The school s student council is composed of 20 members divided among the three classes. Determine a fair number of seats on the council for each class. Each group will consider their assigned fair-division scenario and present their solutions to the class.

9 Fair Division can be DISCRETE or CONTINUOUS. Discrete Division - occurs when the objects of the division cannot be meaningfully separated into pieces. Ex) Dividing a house a piece of a home is not useful!

10 Fair Division can be DISCRETE or CONTINUOUS. Continuous Division - occurs when the objects of the division can be separated into pieces. Ex) Dividing a cake infinite amount of ways!

11 Fair Division can be DISCRETE or CONTINUOUS. It can be difficult to define fairness in some situations because different people place different values on the same object. Today, we ll look at the Continuous Case.

12 Fair Division: The Continuous Case Assume you have 10 pieces of candy. How do you divide them fairly among 5 children? At face value, this question looks simple. However, there are some additional questions that need answering: Are the candies identical? Are the children entitled to equal shares?

13 If the candies are identical and the children are entitled to equal shares, then the problem is simple. Each child gets two candies. If, on the other hand, the answer to either of the above questions is no, the solution is problematic. Should we, regardless of the type of candies involved and regardless of the children s relative entitlement, give each child two pieces of candy anyways?

14 If we did, can we with assurance say we are dividing the candies fairly? What does fairly mean in this context.

15 What if....the candies are replaced by multi-million dollar paintings, jewelry, piece of land, seats in congress

16 So What is a fair share? A fair share is any share that, in the opinion of the person getting it, is at least one Nth of the items to be divided where N is the number of players. The fair share is relative to the one receiving it. Anyone else s opinion doesn t matter. This implies only proportional shares, not envy free shares. Envy Free Shares: each person should feel that the received portion is at least as big as every other person s.

17 4 Fair Divisions: Continuous Case Divider Chooser 2 players Lone Divider 3 players + Lone Chooser 3 players+ Last Diminisher 4 or more players

18 Proportional Methods: 1. Two Players - Divider Chooser One of the players divides the item into two (in his/her opinion) equal parts in a way that he/she thinks is a 50:50 share. The other player then chooses. Hence the name divider-chooser. If played honestly, this procedure tends to be proportional and envy-free. Under this method, both players end up getting shares that, in their opinion, are worth at least one-half of the total

19 Divider Chooser (example) Both players decide who will be the divider and who will be the chooser. Player one (the divider) divides the item into two considerably equal pieces Player one: Divider

20 Divider Chooser (example continued) Player two (the chooser) chooses the piece that he/she wants, and player one receives the leftover piece Player two s claimed piece Leftover piece -> player one s piece

21 2. Three or more players - Methods that are not envy free A.Lone Divider: An extension of the divider-chooser method for more than 2 players. The divider is selected randomly. This fair division scheme requires three steps: Step 1: Divider The divider slices the item into three pieces, each of which is (in his opinion) worth exactly one-third of the total. Step 2: Declarations Each chooser declares which pieces he or she considers acceptable.

22 Step 3: Distribution What happens depends on the declaration. Case 1 one chooser declares more than one piece acceptable. Then the other chooser gets his chosen piece, the chooser of more than one piece gets his other choice and the divider gets the remaining piece. Case 2 Both choosers declare just one piece and they are different. Case 3 Both choosers declare just one piece and it is the same piece. Here, we give the divider one of the two undeclared pieces. Then we put the remaining two pieces back together and apply the divider-chooser method.

23 Lone Divider (example) Divider is chosen randomly Divider divides into 3 pieces Randomly chosen divider

24 Lone Divider (example continued) Players declare which ones they consider desirable Two choosers can either declare one or two pieces Player two Player three Player three Player two

25 Lone Divider (example continued) Since player two and three both declared a piece that the other did not declare they will be given those pieces. The divider will get the leftover piece Player two Player three Player one (divider)

26 Lone Divider Continued If both choosers (player 2 and 3) choose the same piece, the divider (player 1) picks a piece. Then the remaining two pieces are put together and players 1 & 2 use Divider-Chooser Method Player One (Divider) Player three Player two

27 Who might feel slighted? Using Lone Divider, who might feel like they did not get a fair share? Player 1 (Divider) Player 2 (First chooser) Player 3 (Next chooser)

28 Lone Chooser: Also a three step process. It can also be extended out to more than three people. A chooser and 2 dividers are chosen at random Step 1: First division The two dividers split the item by the divider chooser method. Step 2: Second division Each divider now divides his part into three parts he considers equal Step 3: Selection The chooser picks one piece from each divider, and each divider keeps whatever they have left.

29 Lone Chooser (example) Two dividers are chosen at random They divide using the Divider Chooser Method Divider two Divider one

30 Lone Chooser (example continued) Dividers divide their parts in three considerably equal pieces Divider two Divider one

31 Lone Chooser (example continue) Chooser claims a piece from each side The two dividers get the remaining pieces on their side Chooser Divider two Chooser Divider one

32 Last Diminisher Method: In this method, everyone is a divider and a chooser. Step 1 Randomly order the players: P 1, P 2, P 3,., P n Step 2 - P 1 cuts a piece Step 3 P 2 either claims a subpiece of P 1 s piece (becomes a diminisher) or passes to stay in contention for a piece of the rest. (P 2 either agrees with P 1 s piece and lets him keep it OR he cuts P 1 s piece down to size and the remaining amount goes back to the group.)

33 Step 4 Case 1: If P 2 becomes a diminisher: The difference between the claims is put back with the rest and P 1 becomes a contender. Case 2: If P 2 passes: Play passes to P 3 Step 5 Each remaining player, in turn, diminishes or passes. When all have played, the last diminisher gets his/her piece and departs. Step 6 The process begins again with one less player. When 2 players remain, use Divider-Chooser

34 Last Diminisher (example) Randomly order the players by number Player one cuts a piece

35 Last Diminisher (example continued) Player two either claims a sub piece of player one s piece, or passes it to the next player who has the same option. If the player claims a sub piece, he/she becomes a diminisher Leftover Sub piece

36 Last Diminisher (example continued) If player two becomes a diminisher, the leftover of player two s piece is put back with the rest of the whole. If player two does NOT claim the piece, it is passed until someone claims it or the last player gets it

37 Last Diminisher (example continued) Process continues until there are two players left Then the two players use the Divider Chooser Method Player 6 Player 5 Player 1 & 6 Player 1 Player 3 Player 4 Player 2

38 Tonight s HW Complete Derivatives Sheet Read Power Point & Finish Today s Notes handout