Question 13.57

27. The Banach-Knaster last-diminisher method is not the only well-known cake-division procedure that yields a proportional allocation for any number of players. There is also one due to A. M. Fink (sometimes called the lone-chooser method). For three players (Bob, Carol, and Ted), it works as follows:

  1. Bob and Carol divide the cake into two pieces using divide-and-choose.
  2. Bob now divides the piece he has into three parts that he considers to be the same size. Carol does the same with the piece she has.
  3. Ted now chooses whichever of Bob’s three pieces that he (Ted) thinks is largest, and Ted chooses whichever of Carol’s three pieces that he thinks is largest.
  4. Bob keeps his remaining two pieces, as does Carol.
  1. Explain why Ted thinks he is getting at least one-third of the cake.
  2. Explain why Bob and Carol each think they are receiving at least one-third of the cake.
  3. Explain why, in general, this scheme is not envy-free.

27.

(a) Ted thinks he is getting at least one-third of the piece that Bob initially received and at least one- third of the piece that Carol initially received. Thus, Ted thinks he is getting at least one-third of part of the cake (Bob’s piece) plus one-third of the rest of the cake (Carol’s piece).

A-32

(b) Bob gets to keep exactly two-thirds (in his own view) of the piece that he initially received and thought was at least of size one-half. Two-thirds times one-half equals one-third. The same argument applies to Carol.

(c) If, for example, Ted thinks the half Carol initially gets is worthless, then Ted may wind up thinking that he (Ted) has only slightly more than one-third of the cake, while Bob has (in Ted’s view) almost two-thirds of the cake. In such a case, Ted will envy Bob.