Refer to the divide-and-choose method for three players, also known as the lone-divider...

Refer to the divide-and-choose method for three players, also known as the lone-divider method, presented in Section 4.1.

Reference:

DIVIDE-AND-CHOOSE METHOD FOR THREE PLAYERS

Three players, X, Y, and Z, are to divide a cake.

STEP 1: Player X (the divider) divides the cake into three pieces of equal value to him or her.

STEP 2: Players Y and Z (the choosers) decide independently which pieces are worth at least one-third of the cake’s value. Such pieces are said to be acceptable to the player.

STEP 3: The choosers announce which pieces they consider to be acceptable.

STEP 4: The next step depends upon which pieces are declared acceptable by Y and Z.

Case A: If at least one piece is unacceptable to either player Y or player Z, then the divider, player X, gets one of those unacceptable pieces. If both player Y and Z can choose different acceptable pieces from the two remaining pieces, they do so. Otherwise, they must put the remaining two pieces back together and use the divide-and-choose method for two players to divide those reassembled pieces.

Case B: If every piece is acceptable to player Y and player Z, then both Y and Z take pieces they consider acceptable. The divider gets whichever piece is left.

Suppose the three people from the previous problem, Joseph, Paul, and Renn, will share the same pizza (half pepperoni and half olive). This time, suppose Joseph prefers pepperoni by a ratio of 2 to 1 over olive, Paul prefers it by a ratio of 1 to 4, and Renn prefers it by a ratio of 3 to 1. Suppose Joseph is the lone divider and splits the pizza into the following three servings. Use points to assign value.

a. Determine how many points represent a fair share according to each man.

b. Joseph is the divider, so he created three servings that he values equally. Verify that he created three servings of equal value.

c. What value would Renn place on each serving created by Joseph, and which serving(s) would be acceptable to him?

d. What value would Paul place on each serving created by Joseph, and which serving(s) would be acceptable to him?

e. Describe how the servings could be allocated, and explain why the division is proportional.

f. Is the final division of servings envy-free? Explain.