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§ 3.4 The Lone-Chooser Method The Lone-Chooser Method (for three players) Preliminaries: One of the players is designated as the chooser, C. The other two players will be dividers D1 and D2. These assignments will be made randomly. The Lone-Chooser Method (for three players) Step 1. D1 and D2 divide S between themselves into two fair shares using the the divider-chooser method. We will call s1 D1‘s piece and s2 D2‘s piece. The Lone-Chooser Method (for three players) Step 1. D1 and D2 divide S between themselves into two fair shares using the the divider-chooser method. We will call D1‘s piece s1 and D2‘s piece s2. Step 2. Each divider divides his/her share into three subshares. The subshares of s1 are s1a , s1b and s1c ; the subshares of s2 are s2a , s2b and s2c. The Lone-Chooser Method (for three players) Step 3. C now chooses one of D1‘s subshares and one of D2‘s subshares. (a) Draw aHomer, possible Lenny first division Homer Example: and by Carl have other than straight vertical cut through the gotten together to abuy a special-edition $6 Lardcenter and indicate which piece Lenny Lad donut. Half of the donut has sprinkles and would choose. the other half is jelly-filled. Suppose that Homer (b) Based on the first division draw is a possible and Lenny are the dividers and Carl the second division that Homer might make. divider. $2.00 $1.00 $4.50 $1.50 (c) Based on the first division, draw a possible Ssecond S LennyJmight make. J division that S J (d) Based on the second divisions describe a possible final fair division of the donut. $5.00 $4.00 (e) Find the dollar value of each share in the eyes of the player receiving it. The Lone-Chooser Method (for N players) Step 1. D1, D2, D3, . . . , DN-1 divide S fairly among themselves. Step 2. Each divider subdivides his/her share into N subshares. Step 3. C chooses one subshare from each divider. Every player ends up with N - 1 subshares. § 3.5 The Last-Diminisher Method The Last-Diminisher Method The Idea: S is divided into two parts--a claimed piece (the C-piece) and the rest (the R-piece). The players are separated into claimants and nonclaimants. During each turn a player may opt to become a claimant by trimming (or diminishing) the C-piece. The Last-Diminisher Method Preliminaries. Players are put in order (P1, P2, P3, . . . , PN) randomly. The game will be played in rounds. In each round we will have one fewer player than in the last and our set of goods to be divided, S, will be smaller. The Last-Diminisher Method Round 1. The first player P1 becomes the first claimant by cutting a share that he/she believes to be worth (1/N )th of S. P2 may now choose to play or pass on the C-piece. P2 will play if he/she thinks the C-piece is worth more than (1/N )th of S. If P2 plays, then he/she will do so by diminishing the C-piece to what he/she feels is a fair share of S. In this case P2 would now be the claimant The Last-Diminisher Method Round 1. (cont’d) The players continue in this manner until everyone has either played or passed. The final claimant, called the Last-Diminisher, keeps the trimmed C-piece. The Last-Diminisher Method Round 2. Last round’s R-piece is our new S, which is to be divided by the remaining N - 1 players. The process described for round 1 is repeated--this time with one fewer player. This means that a fair share of (the new) S is now 1/(N - 1)th of S. Round 3, 4, . . . Repeat the process, with one fewer player in each round, until only two players remain. Then use Example: Flight 815 has crashed on a strange island in the Southern Pacific. Jack, Kate, Sawyer, Locke and Sayid have decided to divide the island amongst themselves using the Last-Diminisher Method. Example: (exercise 37, pg 124) An island is to be divided among seven players (P1, P2, P3, . . . , P7) using the last-diminisher method. P3 gets his fair share at the end of round 1, and P7 gets her fair share at the end of round 3. There are no diminishers in rounds 2, 4 and 5. (a) Who is the last diminisher in round 1? (b) Which player gets a fair share at the end of round 2? (c ) Which player cuts at the beginning of round 3? (d) Which player gets a fair share at the end of round 4? (e) Which player gets a fair share at the end of round 5?