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.