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:
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).
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(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.