This is in contrast to proportional division procedures, which give each partner at least
of the cake, but may give more to some of the partners.
Moreover, it is possible to divide the cake to any number k of pieces which both partners value as exactly 1/k.
Hence, it is possible to divide the cake between the partners in any fraction (e.g. give 1/3 to Alice and 2/3 to George).
, the division is neither exact nor envy-free, since each partner only values his own piece as
The main mathematical tool used by Austin's procedure is the intermediate value theorem (IVT).
partners who want to divide a cake such that each of them gets exactly one half.
For the sake of description, call the two players Alice and George, and assume the cake is rectangular.
A single knife can be used to achieve the same effect.
Alice must of course end the turn with the knife on the same line as where it started.
Again, by the IVT, there must be a point in which George feels that the two halves are equal.
As noted by Austin, the two partners can find a single piece of cake that both of them value as exactly
, the two partners can divide the entire cake to
for both of them:[2] Two partners can achieve an exact division with any rational ratio of entitlements by a slightly more complicated procedure.
with the Fink protocol, it is possible to divide a cake to
, the generated division is not exact, since a piece is worth
only to its owner and not necessarily to the other partners.
partners; only near-exact division procedures are known.