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**THE MATHEMATICS OF SHARING: FAIR-DIVISION GAMES**

Notes 11 – Section 3.1

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Essential Learnings Students will understand and be able to determine the basic elements of fair-division games. Students will understand fair shares and fair- division methods.

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Fair Division You have 4 children and 20 pieces of candy. How should you divide the candy up? Is that a fair division if the candy pieces are not the same (packs of gum, mini-hershey bars, etc.)?

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**Basic Elements of Fair-Game Division**

The goods (booty) – the item or items being divided (such as candy, pizza, art, land, etc.). The symbol S will be used to denote the goods. The players – the set of parties with a right to share S (this can be individuals or institutions). The value systems – each player can quantify the value of the goods or any of its parts (i.e. to me, that piece is worth 30% of the total value of S).

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Basic Assumptions Rationality – each player is a thinking, rational entity seeking to maximize his or her share of the booty S. A player’s moves are based on reason. Cooperation – the players are willing participants and accept the rules as binding. There are no outsiders such as judges or referees – just the players and the rules.

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Basic Assumptions Privacy – players have no useful information on the other players’ value system or do not know what moves the they are going to make. Symmetry – players have equal rights in sharing the set S. Each player is entitled to a proportional share of S.

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Fair Shares Suppose that s denotes a share of the booty S and that P is one of the players in a fair-division game with N players. We will say that s is a fair share to player P if s is worth at least 1/Nth of the total value of S in the opinion of P. Such a share is often called a proportional fair share, but for simplicity we will refer to it just as a fair share.

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Example Suppose we have 4 players and that S has been divided into four shares: s1, s2, s3, s4. Paul values s1 at 12% of S, s2 at 27% of S, s3 at 18% of S, and s4 at 43% of S. s2 and s4 are both fair shares to Paul, since the threshold for a fair share when dividing among 4 players is 25%.

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**Fair-Division Methods**

A fair-division method is a set of rules that define how the game is to be played. Continuous fair-division game – the set S is divisible infinitely many ways, and shares can be increased or decreased by arbitrarily small amounts. Examples – land, cake, pizza)

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**Fair-Division Methods**

Discrete fair-division game – the set S is made up of objects that are indivisible like paintings, houses, cars, boats, jewelry. Mixed fair-division game – some of the components are continuous and some are discrete. Example – dividing an estate consisting of jewelry, a house, and a parcel of land.

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**Example – Sharing Pizza**

Homer and Marge jointly buy a half pepperoni and half mushroom pizza. Marge values the mushroom pizza three times as much as she values the pepperoni pizza. Find the dollar value to Marge of each of the follow pieces of pizza: The pepperoni half of the pizza

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**Example – Sharing Pizza**

Homer and Marge jointly buy a half pepperoni and half mushroom pizza. Marge values the mushroom pizza three times as much as she values the pepperoni pizza. Find the dollar value to Marge of each of the follow pieces of pizza: The mushroom half of the pizza

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**Example – Sharing Pizza**

Homer and Marge jointly buy a half pepperoni and half mushroom pizza. Marge values the mushroom pizza three times as much as she values the pepperoni pizza. If a slice of mushroom has an angle of 72, find the dollar value to Marge of one slice of mushroom.

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**Example – Sharing Pizza**

Homer and Marge jointly buy a half pepperoni and half mushroom pizza. Marge values the mushroom pizza three times as much as she values the pepperoni pizza. If a slice of pepperoni has an angle of 54, find the dollar value to Marge of one slice of pepperoni.

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Assignment p. 103: 1, 3, 7, 8, 9

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