Last diminisher
The last diminisher procedure is a procedure for fair cake-cutting. It involves a certain heterogenous and divisible resource, such as a birthday cake, and n partners with different preferences over different parts of the cake. It allows the n people to achieve a proportional division, i.e., divide the cake among them such that person receives a piece with a value of at least 1/n of the total value according to his own subjective valuation. For example, if Alice values the entire cake as $100 and there are 5 partners then Alice can receive a piece that she values as at least $20, regardless of what the other partners think or do.
During World War II, the Polish-Jewish mathematician Hugo Steinhaus, who was hiding from the Nazis, occupied himself with the question of how to divide resources fairly. Inspired by the Divide and choose procedure for dividing a cake between two brothers, he asked his students, Stefan Banach and Bronisław Knaster, to find a procedure that can work for any number of people, and published their solution.
This publication has initiated a new research topic which is now studied by many researchers in different disciplines; see fair division.
This is the description of the division protocol in the words of the author:
Each partner has a method that guarantees that he receives a slice with a value of at least 1/n. The method is: always cut the current slice such that the remainder has a value of 1/n for you. There are two options: either you receive the slice that you have cut, or another person receives a smaller slice, whose value for you is less than 1/n. In the latter case, there are n-1 partners remaining and the value of the remaining cake is more than (n-1)/n. Hence by induction it is possible to prove that the received value is at least 1/n.
The last-diminisher protocol is discrete and can be played in turns. In the worst case, actions are needed: one action per player per turn.
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