Refer to the last-diminisher method presented in Section 4.1. THE LAST-DIMINISHER M...

Refer to the last-diminisher method presented in Section 4.1.

THE LAST-DIMINISHER METHOD OF FAIR DIVISION

Suppose any number of players X, Y, . . . are dividing a cake. To give one player a piece of cake that the player considers a fair share and that no other player considers to be more than fair, they will proceed as follows.

STEP 1: Player X cuts a piece of the cake that he or she considers to be a fair share.

STEP 2: Each player in turn judges the fairness of the piece of cake.

Case A: If a player considers the piece to be a fair share or less than a fair share, then it is the next player’s turn to judge the fairness of the piece.

Case B: If a player considers the piece to be larger than a fair share, then that player trims the piece to a smaller size that he or she feels is a fair share. The trimmed-off piece is reattached to the main body of the cake, and it is the next player’s turn to judge the fairness of the just-trimmed piece.

STEP 3: The last player who trimmed the cake to a smaller size gets the piece. If no player trimmed the cake, then player X, who originally cut the piece, gets the piece.

Repeat: After one player takes a piece of cake, begin the whole process again without that player and that piece. When only two players are left, they use the divide-and-choose method for two players to divide the remaining cake fairly.

Describe a situation in which the last-diminisher method leads to a proportional division that is not envy-free.