§ 3.1 Fair-Division

Fair-Division Games Every fair-division game is made up of: 1. A set of goods to be divided. We will refer to this set as S. 2. A set of players : P1 , P2, P3 , . . . , , PN . Each player has his or her own value system.

Fair-Division Games Given the goods and the players, our goal is to end up with a fair division of S--in other words, we wish to divide S so that each player gets a fair share. In this chapter we will consider different sets of rules called fair-division methods.

Fair-Division Games In each fair-division game we consider we will make the following assumptions about the players: Cooperation: Players agree to follow and accept the rules of the game. The game will end with a division of S after a finite number of moves. Rationality: Players act rationally--their value systems conform to the basic arithmetic laws. Privacy: Players have no ‘inside’ information on the other players’ upcoming moves. Symmetry: Players have equal rights in sharing S.

Fair-Division Games If these assumptions hold then a fair-division method is guaranteed to give every player an opportunity to get a fair share of S.

Fair-Division Games Consider a share s of S and a player P. If there are N players then a fair share to player P is a share that is worth at least (1/N )th of the total value of S to player P.

Fair-Division Games There are three types of fair-division game: 1. Continuous: The set S is divisible in infinitely many ways. 2. Discrete: The set S is made up of indivisible objects. 3. Mixed: Some objects in S are continuous while others are discrete.

Example: (exercise 1, pg 112) Alex buys a chocolate-strawberry mousse cake for $12. Alex values chocolate 3 times as much as he values strawberry. (a) What is the value of the chocolate half of the cake to Alex? (b) What is the value of the strawberry half of the cake to Alex? (c ) A piece of the cake is cut as shown in (ii). What is the value of the piece to Alex?

Example: (exercise 5 pg 112) Three players (Ana, Ben and Cara) must divide a cake among themselves. Suppose the cake is divided into 3 slices s1, s2 and s3. The values of the entire cake and of each of the 3 slices in the eyes of each of the players are shown in the following table. (a) Indicate which of the three slices are fair shares to Ana. (b) Indicate which of the three slices are fair shares to Ben. (c) Indicate which of the three slices are fair shares to Cara. Whole Cake s1 s2 s3 Ana $12.00 $3.00 $5.00 $4.00 Ben $15.00 $4.50 $6.50 Cara $13.50

§ 3.2 The Divider-Chooser Method

The Divider-Chooser Method The Idea: “You cut -- I choose.” Given two players, one player is designated as the divider and the other is said to be the chooser. The divider divides the set S into two pieces. The chooser selects the piece he or she wants. (Read example 3.1 on page 91)

§ 3.3 The Lone-Divider Method

The Lone-Divider Method (for three players) Preliminaries: One of the players is designated as the divider, D. The other two players will be choosers C1 and C2. These assignments will be made randomly.

The Lone-Divider Method (for three players) Step 1. The divider D divides S into three pieces: s1, s2 and s3. Step 2. C1 declares which of the three pieces are fair to him/her. C2 does the same independently. Each player must bid for a piece that they feel is worth 1/3 or more of S.

The Lone-Divider Method (for three players) Step 3. The pieces are distributed. How? This depends on the bids. We will separate the pieces into groups: C-pieces (“chosen” pieces that are listed in one of the chooser’s bids) and U-pieces (unwanted pieces that neither chooser wants). There are two cases to consider. . .

The Lone-Divider Method (for three players) Step 3. Case 1. There are at least two C-pieces. Give each chooser one of the pieces they bid for and give the divider the remaining piece. (Players may swap pieces after this if they desire.)

The Lone-Divider Method (for three players) Step 3. Case 2. There is only one C-piece. This means that there are two U-pieces--one is given to the divider. The other U-piece is recombined with the C-piece to make a single big piece called the B-piece. The B-piece is now divided by the two chooser using the divider-chooser method.

Example: Macedonia, 323 BCE Example: Macedonia, 323 BCE. Alexander the Great has just died and his empire is to be divided by the three generals Antigonus, Ptolemy and Salauceus. Antigonus is randomly picked to be the divider. He cuts the empire into three pieces: Greece/Macedonia (s1), Egypt (s2) and Asia (s3). s1 s2 s3 Antigonus 33 1/3 % Ptolemy 25% 55% 20% Salaceus 5% 60% 35% Rome Athens Carthage

Example: Macedonia, 323 BCE Example: Macedonia, 323 BCE. Alexander the Great has just died and his empire is to be divided by the three generals Antigonus, Ptolemy and Salauceus. Antigonus is randomly picked to be the divider. He cuts the empire into three pieces: Greece/Macedonia (s1), Egypt (s2) and Asia (s3). The following bids would be entered: Ptolemy: { s2 } Salaceus: { s2 , s3 } s1 s2 s3 Antigonus 33 1/3 % Ptolemy 25% 55% 20% Salaceus 5% 60% 35% Rome Athens Carthage

The Lone-Divider Method (for more than three players) Step 3. Case 1. There is a way to give each chooser one of the shares listed in his or her bid. The divider gets the last unassigned share. Players may swap pieces after this is done.

The Lone-Divider Method (for more than three players) Step 3. Case 2. There is a standoff. A standoff occurs when more than one chooser is bidding on the same share, or three chooser are bidding on just two shares or if K choosers are bidding on less than K shares. When this happens we first set aside the shares and players involved in the standoff from those that are not.

The Lone-Divider Method (for more than three players) Step 3. Case 2. (cont’d) The players not in the standoff are assigned their shares and quit playing. All the shares that are left are recombined into the whole group and the process is done all over again.

Example: (exercise 16, pg 117) Four partners (Childs, Choate, Chou and DiPalma) want to divide a piece of land fairly using the lone-divider method. Using a map, DiPalma divides the land nto four parcels (s1 ,s2 ,s3 ,s4 )and the choosers make the following declarations: Childs: { s2 ,s3 } Choate: { s1 ,s3 } Chou: { s1 ,s2 } (a) Describe a fair division of the land.

Example: (exercise 21, pg 118) Six players want to divide cake fairly using the lone-divider method. The divider cuts the cake into six slices (s1 ,s2 ,s3 ,s4 , s5 ,s6 ) and the choosers make the following declarations: Childs: {s2 ,s3 } Choate: {s1 ,s3 } Chou: {s1 ,s2 } (a) Describe a fair division of the cake.