The Brams–Taylor method creates an envy-free, fair division involving four people. Suppo...

The Brams–Taylor method creates an envy-free, fair division involving four people. Suppose we want to divide a cake among four people. In the Brams–Taylor method, the divider cuts the cake into more pieces than there are players.

STEP 1: Player A cuts the cake into five pieces he or she believes to be fair. All pieces are passed to player B.

STEP 2: Player B trims at most two of the five pieces to create a three-way tie for the greatest-valued piece. All trimmings are set aside, and the pieces are passed to player C.

STEP 3: Player C trims at most one of the five pieces to create at least a two-way tie for the greatest-valued piece. The trimming is set aside, and the pieces are passed to player D.

STEP 4: Player D chooses a piece from the five that he or she values the most.

STEP 5: Player C chooses a piece from the remaining four that he or she values the most. However, if he or she trimmed a piece, and it is still there, he or she must choose it.

STEP 6: Player B chooses a piece from the remaining three that he or she values the most. However, if he or she trimmed a piece, and it is still there, he or she must choose it.

STEP 7: Player A chooses a piece that has not been trimmed from the remaining pieces.

Notice, at this point, that this method leaves a fifth piece and the trimmings undistributed. One way to handle the extra cake is to simply reapply the method with the understanding that there will again be a fifth piece and trimmings undistributed. The procedure could be repeated over and over until the crumbs are so small that no one would care what became of them. An alternative to this repetitive process is to use a variation of the continuous envyfree division method, described in this section, for the division of the excess.

a. Consider the continuous envy-free division method as described in this section. Pay attention to how the excess is divided and how the second divider and second chooser are determined. When there are four players, explain how the order of cutting and choosing could be shuffled to begin to divide the excess.

b. Think about the four-player situation and the order of cutting, trimming, and choosing as you explain why there must be an extra piece cut at the beginning.

c. Think about the four-player situation and the order of cutting, trimming, and choosing as you explain why player B may trim only two pieces and player C may trim only one piece.