lastdiminisher

The last diminisher is a cake-cutting protocol used to divide a divisible good (the cake) among n players who may have different valuations of the cake. It is designed to produce a proportional division, meaning each player receives a piece they value at least 1/n of the whole according to their own valuation.

The cake is allocated in rounds. With k remaining players, the current piece is initially the whole cake. In turn, each player trims the piece to what they value as at least 1/k of the total. If a trimming occurs, that player becomes the current last diminisher; subsequent players may trim again, replacing the last diminisher. When a complete pass occurs with no new trim, the last diminisher takes the piece, that piece is removed, and the procedure repeats with the remaining k-1 players.

This method guarantees a proportional allocation: the recipient of each piece values it at least 1/k of the original cake, where k is the number of players still in the round. The procedure requires only local information about a single threshold (1/k) and does not require expressing entire valuations. It does not generally ensure envy-freeness.

Historically, the last diminisher is one of the standard proportional division procedures in cake-cutting theory. It is discussed alongside other methods such as divide-and-choose and cut-and-choose and illustrates how sequential trimming and last-winner rules can produce fair allocations.