Transcript How to cut a piece of cake? - William & Mary Mathematics

Lau Ting Sum Samson
Suen 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
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? NO
 Envy-free => Proportional? YES

For 2 players
 One player divides and the other player
chooses
 Proportional? YES
 Envy-free? YES



1.
2.
3.
4.
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.
Proportional? YES
 Envy-free? NO

Problem: Some pieces may consist of tiny
small pieces.
 Solution: Moving Knife Version of Last
Diminisher


1.
2.
3.
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.
Advantage: avoid cutting a piece of cake into
tiny small pieces.
 Proportional???
 Envy-free???

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.

3.
4.
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.)
5.
6.
7.
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.
Proportional? YES
 Envy-free? YES

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

1.
2.
3.
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.