• candy/cake/pizza
• estate (car, house, jewelry, art)
• land (Bosnia-1995)
• divorce settlements

1. goods to be divided (the loot)
2. players--the parties involved
• must be rational
• have value system
• don't know other players' value systems
• willingness to follow rules of the game and play "fair"
3. internal--work without outside help (judge, lawyer, mom)

• Q1: What does it mean for a player to get a fair share?

• Q2: Is it possible to divide the goods such that each player receives a fair share? If so, how?

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• Q1 answer:
  • Given N players, a fair share means any piece that, in the opinion of the player, has value at least one Nth of the total value of the goods

  • This is recipient dependent, i.e. it only matters what the player receiving the share thinks of it. Opinions of other players are irrelevant.
    
    

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• Q2 answer:
  • the rest of the chapter

• continuous:
  • the good are infinitely divisible, can be increased/decreased in arbitrarily small amounts

• discrete:
  • goods are indivisible (paintings, boats, etc.)

• Two players: Divider-Chooser Method
  (you cut, I choose method)

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-coin toss determines divider

-divider cuts "cake" into two "equal" pieces

-chooser selects piece and divider gets the leftover

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Note:
This method guarantees players will get what they think is at least one half of the total share. The divider guarantees this in the act of dividing and the chooser by picking the "biggest" piece.

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• More than two players

• Lone-Divider Method

Using this method, we have one divider and the remaining people are choosers. For simplicity, take three players D=divider; C₁,C₂ ==>choosers.

Steps:

1. (division) Divider cuts cake into three pieces s₁, s₂, s₃. D must cut them equal (in his eyes) because he doesn't know which he will end up with.
2. (bids) Each chooser writes on a piece of paper every share he thinks is fair, in this case at least 1/3 of the cake. These 'bids' must be independent. (It is logically impossible for a chooser to think all three pieces are unfair and thus must bid for at least one piece.)
3. (distribution) Separate pieces into two groups: "bid-for" and "unbid" pieces.
• Case 1: Two or more pieces are in the "bid-for" group. It is easy to give each chooser a piece they bid for and the divider the remaining piece. Each player has a fair share.
• Case 2: Only one piece is in the "bid-for" group, call it C. First we combine C with an "unbid" piece, call the new big piece B. The remaining piece goes to the divider, to whom all pieces were fair. Now we take B and use the divider-chooser method to give each chooser a fair share of the cake.

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• Lone-Chooser Method

Using this method, we have one chooser and the remaining players are dividers. Again for simplicity, take three players where C is the designated chooser and D₁, D₂ are the dividers.

Steps:

1. (first division) D₁ and D₂ cut the cake fairly using divider-chooser method. Each considers his slice worth at least one-half of the total.
2. (second division) Each divider then divides his piece into three equal shares.
3. (selection) The lone chooser picks one piece from each divider, the remaining pieces being kept by the divider. Each person received a fair share of the cake.

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• Last-Diminisher Method

The basic idea of this method is that at any given time, the cake is divided into two pieces: the C-piece and the R-piece. The players are also in two groups: one person is the claimant of the C-piece and the others are the nonclaimants.

Steps:

1. (preliminary)
   Before the game, players are assigned random order to be followed throughout the game. The game is played in rounds, and at the end of each round there is one less player and a smaller cake.
2. (round 1)
   The first player, P₁, begins as the claimant. He cuts a piece off the R-piece that he believes to be exactly one Nth of the total. This share is the C-piece. P₁ doesn't know if he will end up with this piece which ensures he cuts it fairly. The next player, P₂, has the option to become the claimant of the C-piece or to pass it on to P₃. P₂ plays only if he thinks the piece is more than a fair share. If this is the case, he must cut off part and put it back with the R-piece; i.e., diminish the C-piece. Next, P₃ either plays or passes and P₄ the same. At the end of the round, the last person to diminish the piece keeps it and is out of the game.
3. (round 2, 3...)
   The R-piece is now the new cake to be divided among the N-1 remaining players. Staying in the assigned order, P₁ again cuts one (N-1)th of the total. The next player claims and diminishes the piece or passes. This continues until there are two players left. The remaining piece is divided using the divider-chooser method.

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