Brams–Taylor–Zwicker procedure

The Brams–Taylor–Zwicker procedure is a protocol for envy-free cake-cutting among 4 partners.[1]^{: 126–128}

The procedure uses a variation of Austin's procedure for two partners and general fractions. That procedure allows two partners to divide an entire cake to ${\displaystyle k}$ pieces, each of which is worth exactly ${\displaystyle 1/k}$ for both of them.

The main procedure works as follows.

A. Use Austin's procedure with ${\displaystyle k=4}$ and partners #1 and #2. So we have 4 pieces that the first two partners believe to be of exactly identical value of 1/4.

B. Partner #3 trims one piece to create a two-way tie for the largest; the partners now choose pieces in reverse order (#4, #3, #2, #1). Either #4 or #3 must take the trimmed piece. This creates an envy-free division for the entire cake less the trimmings (This is similar to the Selfridge–Conway discrete procedure).

C. Now the trimmings are divided. Assume w.l.o.g. that #3 took the trimmed piece. We use Austin's procedure again with the trimmings and partners #4 and #1, to create 4 pieces each of which equals exactly 1/4 for both of them. Since partners #1 and #2 have an irrevocable advantage over whoever partner that took the trimmed piece, we can let #3 be the first to select a piece from the trimming, then #2, then #4 and #1.