Fair Division

Introduction

Fair DivisionThe Mathematics of Sharing.

We learn to share at a very young age. We share toys, treats and attention. As we get older, we learn to share duties, responsibilities, and blame.

Sharing is a very social interaction. It is important to know how to share using reason and logic rather than bullying our way to a solution.

Fair Division

How can three children divide 27 pieces of candy fairly?

Should they get nine pieces each?

What if there are different types of candy?

Fair Division

I’m allergic to

chocolate

Is this fair?

Is it fair now?

Fair DivisionI’m

allergic to peanut butter

Is this fair?

Is it fair now?

Fair DivisionI like the

light colors the best.

Is this fair?

Is it fair now?

Fair DivisionBut they are only children and it is just candy. Right?

Not exactly!In a divorce or an inheritance proceeding, family heirlooms, land, works of art, and other items of real value are divided.

On a global scale, what about dividing a whole country? (Like the former Yugoslavia in the 1990’s.)

How about dividing the global responsibilities for reducing greenhouse gas emissions?

Candy and cake are convenient representations of items of real value that may need to be divided fairly.

Fair DivisionDefinitions:

The goods (or “booty”): “ S “The item or items being divided.

The shares: “ s1, s2, s3, …. sN”The shares (pieces) of the booty being distributed.

The players: “ P1, P2, P3, …, PN”The set of parties entitled to share the booty.

Fair DivisionDefinitions:

The Value Systems: We will assume that each player has an internalized value system that gives the ability to quantify the value of the booty or any of its parts.

A player can assign a value to each share in absolute terms (“that item is worth $4.50 to me”) or in relative terms (“to me, that item is worth twice as much as the other item”)

Fair DivisionAssumptions:

Rationality: Each player is a thinking rational agent seeking to maximize his/her share of the booty, S. A player’s moves are based on reason alone. (players will not use emotion, mind games, or politics)

Cooperation: The players are willing participants and accept the rules of the game. The game ends after a finite number of moves with a division of S. No outsiders such as judges, consultants, or referees are involved; just the players and the rules.

Fair DivisionAssumptions:

Privacy: Players will have no useful information about the other players’ value systems. (This is sometimes impossible)

Symmetry: All players have equal rights in sharing the set S. If there are two players, each is entitled to a minimum of half of S. If there are three players, each is entitled to at least a third of S.

Fair DivisionFair-Division Scheme: “The rules of the game” Given the booty S, and players P1, P2, P3, …, PN, each with his or her own value system, the ultimate goal is to end up with a fair division of S. The fair-division scheme is the set of rules used to divide S into N equal shares and assign the shares to the players in such a way that each player gets a fair share.

Fair Division

Fair Share: Suppose that s denotes a share of the booty S and 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/N th of the total value of S in the opinion of P.

Definition:

s1

s4

s3

s2

Fair DivisionExample: Suppose that we have four players and a parcel of land has been divided into four shares s1, s2, s3, and s4. Wendy (one of the four players) values the parcels as shown.In this situation, s2 and s4 are both fair shares to Wendy, since the threshold for a fair share when dividing amoung four players is 25% (1/4) of S. Clearly, Wendy would be most happy with s4, but she may have to settle for s2.

43% of S18% of S

27% of S

Wendy’s Values12% of S

Fair DivisionTypes of fair-division problems:

Continuous: A fair-division game in which the booty S, is divisible in infinitely many ways. (land, a cake, pizza, etc.)

Note: When we do cut items such as cake, we will assume that there are no crumbs left as waste.

Fair DivisionTypes of fair-division problems:

Discrete: A fair-division game in which the booty S, is made up of objects that are indivisible. (paintings, cars, houses, hard candy, jewelry, etc.)

Mixed: A fair-division game in which the some of the components of the booty S, are continuous and some are discrete.