Proportional cake-cutting
A proportional cake-cutting is a kind of fair cake-cutting. It is a division of a heterogeneous resource ("cake") that satisfies the proportionality criterion, namely, that every partner feels that his allocated share is worth at least 1/n of the total.
Two assumptions are usually made when proportionality is discussed:
For two people, divide and choose is the classic solution. One person divides the resource into what they believe are equal halves, and the other person chooses the "half" they prefer. The non-atomicity assumption guarantees that the cutter can indeed cut the cake to two equal pieces; the additivity assumption guarantees that both partners value their pieces as at least 1/2.
There are many ways to extend this procedure to more than 2 people. Each way has its own advantages and disadvantages.
Last diminisher is the earliest proportional division procedure developed for n people:
By induction, it is possible to prove that each partner following the rules is guaranteed to get a value of 1/n, regardless of what the other partners do. This is a discrete procedure that can be played in turns. In the worst case, actions are needed: one action per player per turn. However, most of these actions can be done on paper; only n − 1 cuts of the cake are actually needed. Hence, if the cake is contiguous then it is possible to guarantee that each piece is contiguous.
Dubins–Spanier moving-knife procedure is a continuous-time version of Last Diminisher.
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