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.

Melody, June, and Rose will share 24 ounces of pecan pie and 12 ounces of cheesecake. Melody and June prefer pecan pie to cheesecake in a 2-to-1 ratio, while Rose prefers pecan pie to cheesecake in a 3-to-2 ratio. Melody creates a serving consisting of 9 ounces of pecan pie and 2 ounces of cheesecake.

a. Determine what point value represents a fair share according to each woman.

b. Verify that Melody created a serving that, to her, represents a fair share.

c. Calculate the point value of the serving according to June and Rose. Explain why neither of them will trim the piece, and describe how the women should proceed to obtain a proportional division of the desserts.

d. By applying the divide-and-choose method to the remainder of the desserts, suppose the women obtain the following servings.

Serving I (Melody): 9 ounces of pecan pie and

2 ounces of cheesecake

Serving II (Rose): 7 ounces of pecan pie and

6 ounces of cheesecake

Serving III (June): 8 ounces of pecan pie and

4 ounces of cheesecake

Explain whether or not the final division is proportional.

e. Explain whether or not the final division is envy free.