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

2. Basic Elements The goods: the item(s) being divided. Notation: S
The players: the set of parties amongst which S is divided
The value systems: Each player has an internalizes value system that gives the player the ability to quantify the value of the goods or any of its parts, ie To me, that’s worth $37

3. Basic Assumptions
Rationality: Each player is a rational entity seeking to maximize his/her share of S. A player’s moves are based on reason alone (no mind games).
Cooperation: The players are willing participants and accept the rules of the game as binding. (No outside judges)

4. More Assumptions
Privacy: Players have no useful info on the other players’ value system and what kinds of moves others will make in the game.
Symmetry: Players have equal rights in sharing S. So, each player is entitled to at least a proportional share of S.

5. The ultimate goal is to end up with a fair division of S, that is, to divide S into N shares and assign shares to players in such a way that each player gets a fair share.
Suppose that s denotes a share of S and that P is one of N players. We say that s is a fair share to player P if s is worth at least 1/N of the total value of S in the opinion of P.

6. Jif Peanut Butter Commercial

7. 2 Players: Divider-Chooser Method
You cut and I choose.
It’s always better to be the chooser!

8. Dividing a Cake We have a cake which is half chocolate, half lemon
You like both kinds of cake equally. I like chocolate but detest lemon. (And we don’t know what the other person’s preferences are.)

9. Is this a fair cake division?
You cut. I will choose.

10. YES! To you, the chocolate is worth 50% and the lemon is worth 50%.
To me, the chocolate is worth 100%, and the lemon is worth 0%.
Your piece is worth 50%. My piece, in my eyes, is worth 66.66%!
Since we both get a 50% or better share, this is a fair division!

11. Other methods for more players
Lone-Divider
Lone-Chooser
Last Diminisher

12. Method of Sealed Bids
Parents leave a cabin, a vintage car, and a Picasso to their children Art, Betty, Carla, and Dave.
Assumptions:
Each player must have enough money to play the game. Each must be prepared to buy some or all of the items, in order to make honest bids.
Each player must accept money as a substitute for any item.

13. Step 1: Bidding
Each of the players makes a bid for each of the items in the estate. Bids are secret.
Art
Betty
Carla
Dave
Cabin
220,000
250,000
211,000
198,000
Car
40,000
30,000
47,000
52,000
Picasso
280,000
240,000
234,000
190,000

14. Each item goes to the highest bidder for that item.
Step 2: Allocation
Each item goes to the highest bidder for that item.
Cabin: Betty
Car: Dave
Picasso: Art
It is possible that one player gets multiple items, or that another player gets none.

15. Step 3: First Settlement
Depending on what items a player gets in the previous step, he/she will owe money to or be owed money by the estate.
Calculate each player’s fair-dollar share: For each player, add their bids and divide by the number of players.

16. Art
Betty
Carla
Dave
Cabin
220,000
250,000
211,000
198,000
Car
40,000
30,000
47,000
52,000
Picasso
280,000
240,000
234,000
190,000
Bid Total
540,000
520,000
492,000
440,000
Fair-dollar share
135,000
130,000
123,000
110,000
If the total value of the items the player gets is more than his/her fair-dollar share, then the players pays estate the difference.
If the total value of the items is less than the fair share, then the player gets the difference in cash.

17. Art: Fair-dollar share is 135,000 and gets the painting worth 280,000
Art: Fair-dollar share is 135,000 and gets the painting worth 280,000. He owes the estate 145,000.
Betty: Fair-dollar share is 130,000 and gets the cabin worth 250,000. She owes 120,000.
Carla: Fair-dollar share is 123,000. She gets no items, so the estate owes her 123,000 in cash.
Dave: Fair-dollar share is 110,000 and gets car worth 52,000. The estate owes him 58,000.

18. Step 4: Surplus
Art and Betty contributed 265,000 to the estate, and Carla and Dave got 181,000. So, there’s a surplus of 84,000!
The surplus is common money that belongs to the estate, and is thus divided equally among the players.
84,000/4 = 21,000

19. Step 5: Final Settlement
Add the surplus money to the first settlement.
-Art: Gets painting and pays the estate
145,000-21,000 = 124,000
-Betty: Gets cabin and pays the estate
120,000-21,000 = 99,000
-Carla: Gets 123,000+21,000 = 144,000
-Dave: Gets car plus 58,000+21,000

20. Homework Read Chapter 4 Finish Sealed Bids Worksheet
In Chapter 3 Exercises: 1, 3, 53