1. Lau Ting Sum Samson Suen Wai

2.  Discuss what fairness is  Describe some methods for fair division: 1. Divide-and-choose 2. Last Diminisher 3. Selfridge-Conway Envy-free Procedure  Applying fair division algorithms to real life problems

3.  Proportional: each player’s strategy guarantees that player a piece of size at least 1/n of the whole in his or her own estimation  Envy-free: each player’s strategy guarantees that a piece he or she considers to be at least tied for largest  Proportional => Envy-free?  Envy-free => Proportional? YES NO

4.  For 2 players  One player divides and the other player chooses  Proportional?  Envy-free? YES

5.  For 3 or more players  Procedures 1. Player 1 cuts a piece that he thinks is of size 1/n and hands it to player 2. 2. Player 2 trims the piece to 1/n (if she thinks it is bigger than 1/n) and pass it to player 3. 3. The process continues and the last player who cuts the piece of cake will get it. 4. The remaining players repeat the above procedures until every players get a piece of cake.

6.  Proportional?  Envy-free? YES NO

7.  Problem: Some pieces may consist of tiny small pieces.  Solution: Moving Knife Version of Last Diminisher

8.  Procedures: 1. A referee moves the knife from left to right over the cake until one player calls “cut”. 2. The player who calls “cut” gets that piece of cake. 3. Repeat steps 1 and 2 until every player gets a piece of cake.

9.  Advantage: avoid cutting a piece of cake into tiny small pieces.  Proportional???  Envy-free???

10.  For 3 players  Procedures: 1. Player 1 cuts the cake into three pieces. 2. Player 2 trims at most one of the three pieces to create at least a two-way tie for largest. Setting the trimming aside, player 2 hands the three pieces (one of which may have been trimmed) to player 3.

11. 3. Player 3 now chooses one piece. 4. Player 2 next chooses, from the two remaining pieces. (If she trimmed a piece in step 2, and player 3 did not choose this piece, then she must now choose it.)

12. 5. Player 1 receives the remaining piece. 6. The player (other than player 1) who receives an untrimmed piece cuts the trimming into 3 pieces. 7. Players choose pieces of the trimming in the following order: the player who receives a trimmed piece of cake, player 1, the player who cuts the trimming in step 6.

13.  Proportional?  Envy-free? YES

14.  What if the players want the smallest piece of cake instead?  Sharing a room among 2 or 3 roommates  Sharing properties of a wealthy man among 4 women

15. 1. Show that Last Diminisher (Moving Knife Version) is (or is not) proportional. 2. Show that Last Diminisher (Moving Knife Version) is (or is not) envy-free. 3. (Extra Credit/Open Question) Suggest another envy-free allocation among 3 players.