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The Lone-Chooser Method

The Lone-Chooser Method. Notes 13 – Section 3.4. Essential Learnings. Students will understand and be able to use the Lone-Chooser Method to divide the goods fairly. The Lone-Chooser Method (3 players).

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The Lone-Chooser Method

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  1. The Lone-Chooser Method Notes 13 – Section 3.4

  2. Essential Learnings • Students will understand and be able to use the Lone-Chooser Method to divide the goods fairly.

  3. The Lone-Chooser Method (3 players) • In thismethod one player plays the role of chooser, all the other players start out playing the role of dividers. • Preliminaries: We have one chooser Cand two dividers, D1 and D2. • We decide who is what by a random draw.

  4. The Lone-Chooser Method (3 players) • Step 1 (Division): D1 and D2 divide S between themselves into two fair shares using the divider-chooser method. • Let’s say that D1 ends up s1 with D2 and ends up with s2.

  5. The Lone-Chooser Method (3 players) • Step 2 (Subdivision): Each divider divides his share into three subshares. • Thus, D1divides s1 into three subshares, which we will call s1a, s1b, and s1c andD2divides s2 into three subshares, which we will call s2a, s2b, and s2c.

  6. The Lone-Chooser Method (3 players) • Step 3 (Selection): The chooser C now selects one of D1’s three subshares andone of D2’s three subshares (whichever he/she likes best). These two subsharesmake up C’s final share. D1 then keeps the remaining two subshares from s1,and D2 keeps the remaining two subshares from s2.

  7. Example – Lone-Chooser Method • David, Dinah, and Cher are dividing an orange-pineapple cake using the lone-chooser method. The cake is valued by each of them at $27, soeach of them expects to end up with a share worth at least $9.Their individual value systems (not known to one another, but available to usas outside observers) are:

  8. Example – Lone-Chooser Method • David likes pineapple and orange the same. • Dinah likes orange but hates pineapple. • Cher likes pineapple twice as much as she likes orange.

  9. Example – Lone-Chooser Method • After a random selection, Cher gets to be the chooser and thus gets to sit outsteps 1 & 2. • Step 1 (Division): David and Dinah start by dividing the cake between themselves using the divider-chooser method. After a coin flip, David cuts thecake into two pieces.

  10. Example – Lone-Chooser Method • Step 1 (Division) continued: Since Dinah doesn’t like pineapple, she will take the share with the most orange.

  11. Example – Lone-Chooser Method • Step 2 (Subdivision): David divides his share into three subshares that in hisopinion are of equal value (all thesame size).

  12. Example – Lone-Chooser Method • Step 2 (Subdivision) continued: Dinah also divides her share into three smaller subshares that inher opinion are of equal value. • Remember that Dinah hatespineapple. Thus, she has made her cuts in such a way as to have one-third ofthe orange in each of the subshares.

  13. Example – Lone-Chooser Method • Step 3 (Selection): It’s now Cher’s turn to choose one subshare from David’s three and one subshare from Dinah’sthree.

  14. Example – Lone-Chooser Method • Step 3 (Selection) continued: Cher will choose one of the two pineapple wedges from David’ssubshares andthe big orange-pineapple wedge from Dinah’ssubshares.

  15. Example – Lone-Chooser Method • Step 3 (Selection): The final fair division of the cake is shown. David gets a final share worth $9, Dinah gets a finalshare worth $12, and Cher gets a final share worth$14. David is satisfied, Dinah is happy, and Cher isecstatic.

  16. The Lone-Chooser Method for N Players • In the general case of N players, the lone-chooser method involves one chooser Cand N – 1 dividers D1, D2,…, DN–1. • As always, it is preferable to be a chooserthan a divider, so the chooser is determined by a random draw.

  17. The Lone-Chooser Method for N Players • Step 1 (Division): D1, D2,…, DN–1 divide fairly the set S among themselves,as if C didn’t exist. • This is a fair division among N – 1 players, so each onegets a share he or she considers worth at least of 1/(N–1)thof S.

  18. The Lone-Chooser Method for N Players • Step 2 (Subdivision): Each divider subdivides his or her share into N sub-shares. • Step 3 (Selection): The chooser C finally gets to play.C selects one sub-share from each divider – one subshare from D1, one from D2, and so on. • At the end,C ends up with N – 1 subshares, which make up C’s finalshare, and each divider gets to keep the remaining N – 1 subshares in hisor her subdivision.

  19. The Lone-Chooser Method • When properly played, the lone-chooser method guarantees that everyone,dividers and chooser alike, ends up with a fair share.

  20. Assignment p. 109: 33 & 34

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