Fair DivisionAll fair division problems share the same two basic elements:

1. A set of goods to be divided. They can be anything that has a potential value.

Typically, the goods are tangible objects, i.e. candy, jewelry, property, etc.

The symbol S will be used to denote the object(s) to be divided.

2. A set of players, who are entitled to share the set S. These will be denoted by

P1, P2, ...Pn. Usually people, but can be countries, groups, companies, etc.Each player must have a value system giving that player the ability to

assignvalue not only to the set S but to any part of it.Key Questions:

1. What does it mean for a player to get a fair share of S?

2. Is it possible to divide S into shares (one for each player) in such a way that every player gets a fair share? If so, how?

1. What does it mean for a player to get a fair share of S?

Fair share will mean any share (piece) that in the opinion of the player receiving it has a value that is at least one Nth of the value of S.

2. Is it possible to divide S into shares (one for each player) in such a way that every player gets a fair share? If so, how?

1. Internal -- don't need a judge or ref to help figure it out

2. Players act in a rational manner, i.e. value systems conform to the basic laws of arithmetic.

3. Players have no knowledge about each other's value system,

don't want opponent to know your likes and dislikes.

4. Willingness of players to abide by the rules and results of the game.

Types of Problems:

Continuous -- the set S is divisible in infinitely many ways and shares can be increased or decreased by arbitrarily small amounts

examples: dividing land, cake, pizza, etc.

Discrete -- the set S is made up of objects that are indivisible.

examples: paintings, houses, jewelry, etc.

Mixed -- the set S is made up of both continuous and discrete objects.

example: dividing an estate, consisting of a house, a car and land.

Two Players: The Divider-Chooser Methoda.k.a. "You cut and I choose"

One player divides the "cake" into two pieces and the 2nd player picks the piece he or she wants. When played honestly, this guarantees that each player will get a share that he /she believes to be worth at least one-half of the total

Example: Half moon cookie

Joey Sue

50%50%

Example: Half moon cookie

I like both chocolate and vanilla

I HATE chocolate

100%0%

Is this a fair division of the cookie?

What's Joey's opinion what's left?

What's Sue's opinion of her piece?

A

B

So to make it fair, Joey and Sue flip a coin to seewho is going to cut the cookie. Joey wins and makes the following cut...and Sue takes side B

What if there is more than two players?

we'll look at three different solutions:

1. The lone-divider method

2. The lone-chooser method

3. The last-diminisher method

One person is the divider, chosen randomly. The remaining players "bid" on which of the pieces cut by the divider are fair shares. The pieces are then separated into two groups: the "bid for" and the "unbid for"

Case 1: The "bid for" group has two or more pieces in it. Give each

chooser the piece they bid on and the divider the last remaining piece.

Case 2: There is only one piece in the "bid for' group. Combine this one

piece with one of the "unbid for" pieces. The remaining piece goes to the divider. The new piece is then divided using the divider- chooser method.

1. The lone-divider method --

Case 1: The "bid for" group has two or more pieces in it. Give each

chooser the piece they bid on and the divider the last

remaining piece.

Bid For:

Un-bid For:

s1

s2

s3

Hector TheoLarry

Case 2: There is only one piece in the "bid for' group. Combine this one

piece with one of the "unbid for" pieces. The remaining piece goes to the divider. The new piece is then divided using the divider-chooser method.

Bid For:

Un-bid For:

s1

s2

s3Hector Theo Larry

Hector Theo

Larry

s3

Bid For:

s1 Un-bid For:

s2

s 1 s 2 s 3

Hector 33.3% 33.3% 33.3%

Theo 35% 10% 55%

Larry 40% 25% 35%

Example #2

Larry

Theo

Hector

cake =$25.00Larry ends up with S1

=40%

Hector likes S 3× more than C,and V. How much is S worth to Hector?

cost = $25.00

p. 96 # 1-4 all

p. 97 # 5-9

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