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Section 3.2. Two players: The Divider-Chooser Method. The best known of all continuous fair-division methods Can be used anytime there is a continuous fair-division problem. Division. Player 1: (The Divider) – divides the cake into two pieces that he feels are fair shares

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Section 3.2

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Section 3.2

Two players:The Divider-Chooser Method


  • The best known of all continuous fair-division methods

  • Can be used anytime there is a continuous fair-division problem


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


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]


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


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


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.


  • 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.


Page 117 # 15

  • What should the division be?

  • A) Chase: Chandra: Divine:

  • B) Chase:Chandra: Divine:

  • C) Chase:Chandra: Divine:

  • D) Chase:Chandra: Divine:


Page 117 # 15

  • What should the division be?

  • A) Chase: S2 Chandra: S1Divine:S3

  • B) Chase:S2Chandra:S1Divine:S3

  • C) Chase:S1Chandra: S2Divine:S3

  • D) Chase: ½ (S1 + S2)

    Chandra: ½ (S1 + S2) Divine: S3


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.


  • 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.


Assignment

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

  • Section 3.3: page 117 – 119 # 16 - 24


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