Refer to the divide-and-choose method for two players presented in Section 4.1. Referen...

Refer to the divide-and-choose method for two players presented in Section 4.1.

Reference:

DIVIDE-AND-CHOOSE METHOD FOR TWO PLAYERS

Two players, X and Y, are to divide a cake.

STEP 1: Player X divides the cake into two pieces that he or she considers of equal value. (We call player X the divider.)

STEP 2: Player Y picks whichever of the two pieces he or she considers to be of greater value. If player Y also considers the pieces to be of equal value, then he or she can pick either piece. (We call player Y the chooser.)

STEP 3: Player X gets the piece that player Y did not select.

Letha and Bruce each have a cellular phone. The company they use offers a plan that lets them allocate minutes to their two phones as they wish. The company offers a total of 700 daytime minutes and 1000 evening minutes. Letha uses her phone during the day for work, so she prefers daytime to evening minutes in a 3-to-1 ratio. She decides to allocate 500 day and 50 evening minutes to one phone and the remaining minutes to the other. Bruce will choose which phone he wants to use. He prefers daytime to evening minutes in a 4-to-3 ratio.

a. What value would Letha place on the minutes allocated to each phone?

b. What value would Bruce place on the minutes allocated to each phone, and which would he choose?

c. Is this a proportional division of the minutes?

Explain.

d. Is this division envy-free? Explain.