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Two players: The Divider-Chooser Method

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1 Two players: The Divider-Chooser Method
Section 3.2 Two players: The Divider-Chooser Method

2 The best known of all continuous fair-division methods
Can be used anytime there is a continuous fair-division problem

3 Division Player 1: (The Divider) – divides the cake into two pieces that he feels are fair shares Player 2: (The Chooser) – picks the piece he wants, leaving the other piece to the divider When played properly, this method guarantees that both players get a share they believe to be fair

4 Jif Peanut Butter Commercial
Jake: Mom it’s the last slice [of bread]. Mom: Hmm…Well then let’s share. We’ll cut it in half. Cody: His half can’t be bigger than mine! Mom: All right. I’ll tell you what. Jake gets to cut. Jake: Yes! [cuts a big piece for himself] Mom: But…Cody gets to choose. Jake: [sad about this new information, passes the plate] Cody: Nice…[after a moment, takes bigger piece for himself]. I got a pretty big half! Jake: [smiles despite ending up with a smaller piece]

5 Page 91: Example 3.1 It is better to be the chooser because the divider always gets a piece that is worth exactly one-half, but the chooser may get a piece worth more than one-half

6 Section 3.3: The Lone-Divider Method
A clever way to extend some of the ideas in the divider-chooser method to the case among three or more players For three players- 1 player is randomly chosen to be the divider and the other two players are to be the choosers C1 and C2

7 The Division (For three players)
Step 1: The Divider cuts the cake into 3 pieces, called S1, S2 and S3. The divider must cut the pieces into what he considers 1/3 of the value of the cake. Step 2: The choosers must declare their bids- they write on a slip of paper which piece or pieces they consider to be a fair share. They can vote on all three, but has to be at least one.

8 Step 3: We distribute the pieces depending on the bids
Step 3: We distribute the pieces depending on the bids. There are 2 cases: Case 1: If C1 and C2 bid on different pieces, then C1 gets a piece, C2 gets a piece and the divider gets the piece that is not taken Case 2: If C1 and C2 bid on the same piece, they choose one of the other pieces to give to the divider. They then combine the other two pieces and use the divider-chooser method.

9 Page 117 # 15 What should the division be? A) Chase: Chandra: Divine:
B) Chase: Chandra: Divine: C) Chase: Chandra: Divine: D) Chase: Chandra: Divine:

10 Page 117 # 15 What should the division be?
A) Chase: S2 Chandra: S1 Divine:S3 B) Chase:S2 Chandra:S1 Divine:S3 C) Chase:S1 Chandra: S2 Divine:S3 D) Chase: ½ (S1 + S2) Chandra: ½ (S1 + S2) Divine: S3

11 For more than three players (N)
Step 1: The divider cuts the cake into N pieces, all of which he considers to be a fair share Step 2: Each chooser makes their bid Step 3: Distribute the pieces Case 1: If every chooser has a different bid, the pieces are given to each chooser and the divider gets the last piece.

12 Case 2: If two or more choosers are all bidding on the same piece, then there is a standoff. The players who are not involved, including the divider, divide the pieces according to their bids. Then the remaining pieces are combined and the process starts over.

13 Assignment Section 3.2: page 115 - 116 # 9, 10, 13, 14

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