Fair Division

1. Fair Division Lone Divider Method

2. Lone Divider method T. Serino The Lone Divider Method is a fair division scheme that can be used to divide a bounty between three or more players. As the name implies, there is only one divider in the method and the rest of the players are choosers (we’ll call them chooser 1, chooser2, etc.).

3. Lone Divider method T. Serino Steps: 1. Decide which player is the divider. (roll a die, draw straws, draw cards, buck up...) 2. The Divider cuts the cake into 3 slices (S1, S2, and S3). The cut must be fair to the divider (Each slice must be worth 1/3 of the whole cake). 3. Each of the choosers declares independently (secretly, usually by writing it down on a slip of paper) which slice(s) they each consider to be fair (worth at least 1/3) 4. The cake is distributed. (If the cake cannot be distributed fairly at this point, other divisions may be necessary)

4. Lone Divider method T. Serino Remember: As in all fair division schemes, the Lone Divider Method guarantees that each of the n players (in this case n=3) will receive what they consider a fair share. A fair share is worth at least 1/n of the whole cake (in this case, 1/3 of the whole cake).

5. Lone Divider method T. Serino Example 1) At least one of the choosers selects more than one slice to be fair. (possible to distribute the cake fairly without further division) Scenario:  The divider cuts the cake and labels the slices as shown:  Chooser 1 declares: {S1, S3}   Chooser 2 declares: {S3}

6. Lone Divider method T. Serino Solution:  Chooser 1 declared: {S1, S3}  Chooser 2 declared: {S3} The divider gets S2 (she declared this as a fair slice by making the cut)  Chooser 2 gets S3  Chooser 1 gets S1

7. Lone Divider method T. Serino Example 2) Each chooser selects one slice only,  but they each select a different slice. (possible to distribute the cake fairly  without further division) Scenario:  The divider cuts the cake and labels the slices as shown:  Chooser 1 declares: {S1}  Chooser 2 declares: {S3}

8. Lone Divider method T. Serino Solution:  Chooser 1 declared: {S1} Chooser 2 declared: {S3} The divider gets S2 (she declared this as a fair slice by making the cut)  Chooser 2 gets S3  Chooser 1 gets S1

9. Lone Divider method T. Serino Example 3) Each chooser selects one slice only,  and they both select the same slice. (Further division is necessary) Scenario:  The divider cuts the cake and labels the slices as shown:  Chooser 1 declares: {S3}  Chooser 2 declares: {S3}

10. Lone Divider method T. Serino Solution:  Chooser 1 declared: {S3} Chooser 2 declared: {S3} The divider gets to choose either S1 or S2. (Each chooser declared S1 and S2 as unfair.) The remaining slices are put back together and the choosers will use divider-chooser method to divide it fairly. 

11. Lone Divider method T. Serino *Important notes for understanding. By selecting S3 as the only fair slice, the choosers are implying that both S1 and S2 are each less than 1/3 of the whole cake. If, the slice taken by the divider is worth less than 1/3 of the cake, then the value of the remaining slices is worth more than 2/3 of the cake. Therefore, the choosers will each be guaranteed more than their fair share of the cake.

12. Lone Divider method T. Serino Example 3) (continued) One possible scenario: The Divider selects S1 S2 and S3 are combined. The choosers use divider-chooser method.

13. Try this. T. Serino 1 If you were the divider in a fair division game with three players, show two different ways that you can make a fair division of the following rectangular cake.

14. Try this. T. Serino 2 If you were one of the choosers, which slices would you declare as fair in each of the following examples? S1 S1 S2 S2 S3 S3

15. Try this. T. Serino 3 The divider has cut a cake as shown below. Make a fair division of the cake given each chooser’s declarations. (Decide who gets which piece) S2 Chooser 1: {S1} Chooser 2: {S1, S3} S1 S3

16. Lone Divider method T. Serino

17. M athematical D ecision M aking