Dividing a cake among three people does not mean everyone measures value the same way. One person may value the chocolate more, another the edge, another the fruit. Fairness is not that the pieces look equal to an outside judge, but that no one prefers someone else's piece.
The Cake No One Wants to Trade
Riddle statement
Three people value different regions of a cake differently.
Is there a procedure, based only on cutting and choosing, that guarantees that in the end no one prefers another person's portion to their own?
Describe the full procedure.
Show solution
Solution
Yes. The Selfridge-Conway procedure guarantees an envy-free division for three people.
Call the three people A, B, and C.
1. Main division
A cuts the cake into three pieces that, according to A, have equal value.
B examines the pieces. If there is one that B considers strictly best, B trims it until, according to B, it ties with the second-best piece. The trimming is set aside. Call the trimmed piece $X$.
C chooses first among the three main pieces.
- If C does not choose $X$, B must choose $X$.
- If C chooses $X$, B chooses whichever of the other two pieces B prefers.
A receives the remaining piece.
At this point, the main cake has been divided without envy:
- C chose first.
- B receives $X$ or a piece that B values at least as much as $X$.
- A receives an original untrimmed piece, which was worth one third according to A's own cut.
2. Division of the trimming
Between B and C, one person received $X$ and the other did not. The person who did not receive $X$ cuts the trimming into three pieces that they consider equal.
The order of choice is:
- the owner of $X$ chooses first;
- A chooses second;
- the person who cut receives the remaining piece.
Thus, the cutter of the trimming receives one of three pieces they consider equal; A chooses before that person; and the owner of $X$ chooses first.
A also does not envy the owner of $X$: before the trimming was removed, A valued A's own piece as much as the original full piece from which $X$ came. The piece $X$ plus any share of the trimming can never exceed that original full piece.
Answer: yes. The procedure above divides the entire cake and guarantees that no one prefers another person's bundle.