The Fink protocol[1] (also known as Successive Pairs[2] or Lone Chooser[3]) is a protocol for proportional division of a cake.
Its main advantage is that it can work in an online fashion, without knowing the number of partners in advance.
When a new partner joins the party, the existing division is adjusted to give a fair share to the newcomer, with minimal effect on existing partners.
Its main disadvantage is that, instead of giving each partner a single connected piece, it gives each partner a large number of "crumbs".
We describe the protocol inductively for an increasing number of partners.
When partner #1 enters the party, he just takes the entire cake.
The new partner chooses the piece that is better in his eyes.
The value of each partner is thus at least 1/2 (just like in the divide and choose protocol).
When partner #3 joins, both partners #1 and #2 cut their share to 3 pieces that are equal in their eyes.
from the piece of #2, which gives him at least 1/3 of the total cake.
In general, when partner #i enters the party, the previous i-1 partners divide their share to i equal pieces and the new partner picks a piece from each pile.
Again it is possible to prove that the value of each partner is at least 1/n of the total, so the division is proportional.