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.

Describe a situation that shows how the divide-and choose method for three players can lead to a fair division that is not envy-free.