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.