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 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.
Equivalently,[citation needed] each partner 1, 2, ..., n−1 in turn cuts a slice from the remaining cake.
Then in reverse order, each partner n, n−1, ..., 1 in turn selects a slice that has not yet been claimed.
On the other hand, it is possible to restrict the cuts in order to guarantee that the pieces have a nice shape.
In particular: A continuous-time version of this protocol can be executed using the Dubins-Spanier Moving-knife procedure.
Similar to the last diminisher procedure, it can be used to cut the cake into contiguous parts for each player.
When there are 3 or more partners, the division obtained by the last-diminisher protocol is not always envy-free.
For example, suppose the first partner Alice receives a piece (which she values as 1/3 of the total).
Then, the other two partners Bob and Charlie divide the remainder in such a way that is fair in their opinion, but in Alice's opinion Bob's share is worth 2/3 while Charlie's share is worth 0.
I.e, a partner that won a piece by being the last diminisher, does not have to leave the game, but may rather stay and participate in further steps.
If he wins again, he must release his current piece and it is returned to the cake.
In order to make sure that the protocol terminates, we select a certain constant
The method is: always cut the current slice such that the remainder has a value of
See envy-free cake-cutting#Connected pieces for other solutions to this problem.