Chapter 3: The Mathematics of Sharing Fair-Division Games

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

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)

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.

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.

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

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.)

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

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!

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

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.

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

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.

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.

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.

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.

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

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

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