Selfridge-Conway
Fair Division Procedure
An Envy-Free Cake Division Procedure
The Selfridge-Conway Procedure
• We consider the case where there are exactly 3 participants.
• Suppose the participants are Alex (A), Barry (B) and Charles (C).
• Suppose A, B and C will divide a cake.
• Suppose all three have equal rights to a fair share of the cake.
• We follow the Selfridge-Conway procedure to divide the cake so that
none of the participants will envy what the other receives.
The Selfridge-Conway Procedure - Outline
• The basic outline of the procedure is as follows:
• A will cut the cake into three pieces that he considers equally fair.
• B will have a chance to examine all three pieces and determine if
there are at least two pieces that are tied for “best” – at least in B’s
point of view.
• Assuming B agrees that there are at least two pieces tied for best,
then C will be the first to choose a piece.
• C chooses any piece, then B choose one (there is still at least one
remaining of those tied for best) and finally A chooses.
The Selfridge-Conway Procedure – Outline Continued
• However, it is possible that when B has the opportunity to examine
all three pieces of the cake, B might decide that there is really only
one piece that is the best (at least from B’s point of view).
• If C were given first choice, B might not get that best piece.
• Therefore, if B believes there is only one “best” piece, then B has the
opportunity to trim that best piece to make sure that there are then at
least two that would be tied for “best piece”.
• The trimmings that B cuts from the best piece are set aside for stage
2 of the process. (So there is a stage 2 only if B trims one piece).
• To finish stage 1, C will choose first from the 3 pieces (not counting
the trimmings). Then B will choose. A requirement of the procedure
is that if B did trim a piece, and if that piece is still available after C
chooses, then B must take that trimmed piece. Finally, A will take the
remaining piece.
The Selfridge-Conway Procedure – Outline Continued
Stage 2 – assuming that B trimmed a piece
Now there are two possibilities:
1. B trimmed a piece and C takes it.
2. B trimmed a piece, C does not take it, and therefore B must take it.
In the first case – Suppose C takes the piece that B trimmed.
Then B will divide up the trimmings and the order of selection is
C chooses first, then A and finally B takes what remains.
In the second case – Suppose B takes the trimmed piece.
Then C will divide up the trimmings and the order of selection is
B chooses first, then A and finally C takes what remains.
Envy-Free
• Selfridge-Conway cake division procedure is
designed to be envy-free. This means when the
division is completed, no participant will envy
what another receives.
• To prove this is true, we show that no participant
experiences envy in either of the two stages of
the process and therefore at no time in the
process.
Stage 1
• A does not experience envy because A was able
to divide the cake in a way that he thought all
pieces were fair.
• B will not envy either of the others because B
was able to make sure that there were at least
two pieces that were tied for best, and B is able
to choose one of those.
• C will not envy either of the others because C is
able to choose first in stage 1
Stage 2
• Stage 2 only occurs if B cut some trimmings from one of the pieces
originally cut by A in stage 1.
• We show that no participant will envy what another receives in this
stage.
• To be complete, we need to show that this is true in both of the two
possible cases that can occur in stage 2
Stage 2
• Consider A first:
• The two cases in this stage are:
– 1. If C took the trimmed piece in stage 1, then B will divide the trimmings
and C selects first, then A and then B
– 2. If C did not take the trimmed piece in stage 1, then B must take it, so
therefore C will divide the trimmings, B selects first, then A and finally B.
• To see A will not experience envy in this stage, we see that A will not
experience envy in either case.
• First, A had cut all three pieces in stage 1 and therefore will not envy
the one who took the trimmed piece. A would not have envied
anyone who had gotten the whole piece, much less, that piece plus
only some of what had been trimmed from that piece. Furthermore,
A will select before the other participant that did not take the trimmed
piece and so won’t envy that participant either.
Stage 2
• Next, consider B:
• Again, the two cases in this stage are:
– 1. If C took the trimmed piece in stage 1, then B will divide the trimmings
and C selects first, then A and then B
– 2. If C did not take the trimmed piece in stage 1, then B must take it, so
therefore C will divide the trimmings, B selects first, then A and finally C.
• In case 1, B will not envy the others in this stage because B will
divide the trimmings to ensure that each piece is sufficiently
appealing. In this case B is dividing the trimmings but choosing last
so must divide fairly so as not to envy what the others select.
• In case 2, B will not envy the others because B is selecting first from
the trimmings.
Stage 2
• Finally, consider C:
• For reference, the two cases in this stage are:
– 1. If C took the trimmed piece in stage 1, then B will divide the trimmings
and C selects first, then A and then B
– 2. If C did not take the trimmed piece in stage 1, then B must take it, so
therefore C will divide the trimmings, B selects first, then A and finally C.
• The argument is equivalent to what was just presented for B
• In case 1, C will not envy the others because C can choose first.
• In case 2, C will have the opportunity to divide the trimmings to
ensure that each piece is equally attractive. C must do this because
C will select last in this stage. So C divides the trimmings so that
each piece is equally good and will not envy what the others
choose.