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 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!

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.

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?

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

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 P if, according to P’s value system, s is worth at least 1/n of the total value of the goods.

Example

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, …