How to cut a piece of cake?

How to cut a piece of cake? Lau Ting Sum Samson SuenWai

Objectives • 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

What is fair? • 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? NO YES

Divide-and-Choose • For 2 players • One player divides and the other player chooses • Proportional? • Envy-free? YES YES

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

Last Diminisher • Proportional? • Envy-free? YES NO

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

Last Diminisher (Moving Knife Version) • Procedures: • A referee moves the knife from left to right over the cake until one player calls “cut”. • The player who calls “cut” gets that piece of cake. • Repeat steps 1 and 2 until every player gets a piece of cake.

Last Diminisher (Moving Knife Version) • Advantage: avoid cutting a piece of cake into tiny small pieces. • Proportional??? • Envy-free???

Selfridge-Conway Envy-free Procedure • For 3 players • Procedures: • Player 1 cuts the cake into three pieces. • 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.

Selfridge-Conway Envy-free Procedure • Player 3 now chooses one piece. • 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.)

Selfridge-Conway Envy-free Procedure • Player 1 receives the remaining piece. • The player (other than player 1) who receives an untrimmed piece cuts the trimming into 3 pieces. • 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.

Selfridge-Conway Envy-free Procedure • Proportional? • Envy-free? YES YES

Further Extensions • 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

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

How to cut a piece of cake?