Austin moving-knife procedures


The Austin moving-knife procedures are procedures for equitable division of a cake. They allocate each of n partners, a piece of the cake which this partner values as exactly of the cake. 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.
When, the division generated by Austin's procedure is an exact division and it is also envy-free. 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.
When, the division is neither exact nor envy-free, since each partner only values his own piece as, but may value other pieces differently.
The main mathematical tool used by Austin's procedure is the intermediate value theorem.

Two partners and half-cakes

The basic procedures involve partners who want to divide a cake such that each of them gets exactly one half.

Two knives procedure

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.

Two partners and general fractions

As noted by Austin, the two partners can find a single piece of cake that both of them value as exactly, for any integer. Call the above procedure :
By recursively applying, the two partners can divide the entire cake to pieces, each of which is worth exactly for both of them:
Two partners can achieve an exact division with any rational ratio of entitlements by a slightly more complicated procedure.

Many partners

By combining with the Fink protocol, it is possible to divide a cake to partners, such that each partner receives a piece worth exactly for him:
Note that for, the generated division is not exact, since a piece is worth only to its owner and not necessarily to the other partners. As of 2015, there is no known exact division procedure for partners; only near-exact division procedures are known.