Lau Ting Sum Samson Suen Wai. Discuss what fairness is Describe some methods for fair division: 1....
-
Upload
annice-warner -
Category
Documents
-
view
214 -
download
1
Transcript of Lau Ting Sum Samson Suen Wai. Discuss what fairness is Describe some methods for fair division: 1....
How to cut a piece of cake?
Lau Ting Sum SamsonSuen Wai
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
Objectives
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?
What is fair?
YESNO
For 2 players One player divides and the other player
chooses Proportional? Envy-free?
Divide-and-Choose
YESYES
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.
Last Diminisher
Proportional? Envy-free?
Last Diminisher
YESNO
Problem: Some pieces may consist of tiny small pieces.
Solution: Moving Knife Version of Last Diminisher
Last Diminisher
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.
Last Diminisher (Moving Knife Version)
Advantage: avoid cutting a piece of cake into tiny small pieces.
Proportional??? Envy-free???
Last Diminisher (Moving Knife Version)
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.
Selfridge-Conway Envy-free Procedure
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.)
Selfridge-Conway Envy-free Procedure
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.
Selfridge-Conway Envy-free Procedure
Proportional? Envy-free?
Selfridge-Conway Envy-free Procedure
YESYES
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
Further Extensions
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.
Homework