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Chapter 3: The Mathematics of Sharing

Chapter 3: The Mathematics of Sharing. Fair-Division Games. “If you want to know the true character of a person, divide an inheritance with him.” – Ben Franklin

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Chapter 3: The Mathematics of Sharing

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  1. Chapter 3: The Mathematics of Sharing Fair-Division Games

  2. “If you want to know the true character of a person, divide an inheritance with him.” – Ben Franklin • There are many situations where we need to divide something (a pizza, cost of rent, an inheritance, seats in Congress, land) among several people in as fair a manner as possible. • This is not always simple!

  3. Main Questions • Given a set of goods to divide between several people who all have an equal right to it, how can we divide the goods fairly? • What does it mean for a division to be fair? • Can we guarantee that everyone gets a fair share? • We can think of this as a game: with rules, players, strategies, and moves. • The goal of the game is for everyone to end up with a fair share.

  4. Components of a Fair-Division Game • Goods (“booty): the objects being divided • E.g. cake, candy, money, jewelry, land, etc. • Could also be negative items: chores, bills, debts, etc. • Players: the people who will share the goods • Value system: Each player has their own internal value system that determines how much each part of the goods is worth to them • E.g. a vegetarian will not put much value on the pepperoni half of a half cheese, half pepperoni pizza • Which would you rather have, chocolate or white cake?

  5. Assumptions • Rationality: • All players want to maximize their share of the goods • Act purely rationally (no emotions, mind games, “psyching out”, etc.) • Cooperation: • All players agree to follow the rules • The game will end after a finite number of steps • Privacy: • Players have no knowledge of the value systems of the other players • Symmetry: • All players have an equal right to the goods

  6. Fair Share • The game will end with a fair division of the goods: each player gets a fair share. • So what is a fair share? • Definition: Suppose P is one player in a fair division game with n players, and s is a share of the goods. s is a fair share to player Pif, according to P’s value system, s is worth at least 1/n of the total value of the goods.

  7. Example

  8. Types of Fair-Division Methods • There are many different fair-division methods (different rules of the game) that can be used to make a fair division. • They can be classified based on the number of players they work for and based on the set of goods: • Continuous: the goods can be divided into arbitrarily small amounts • EX: cake, pizza, land, … • Discrete: the goods are a collection of objects that cannot be divided • EX: cars, jewelry, pieces of candy, furniture, …

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