# Brams-Taylor Envy-Free Method of Fair Division

The key to this idea is for the designated cutter to cut the cake into more pieces than there are players. Let's look at it with four players. Let's say the players are : Almaz, Birur, Dehab, and Lul. Let's also say that Almaz is the designated cutter.

## The method

1. Almaz cuts the cake into five (one more than there are players) pieces she thinks to be of the same size. She passes all the pieces to Birur.

2. Birur trims at most two of the five pieces so as to create at least a three-way tie for the largest. He sets the trimmings aside, and passes all the pieces to Dehab. Note that one or two pieces might have been trimmed.

3. Dehab trims at most one of the five pieces so as to create at least a two-way tie for the largest. She might have to trim a piece that was already trimmed by Birur in the above step. She sets the trimmings aside, and passes all the pieces to Lul. Note that one or no pieces might have been trimmed.

4. Lul now chooses from among the five pieces, one that she thinks to be at least tied for the largest piece. She, if she wishes so, might pick a trimmed piece.

5. Dehab chooses next, from among the remaining four pieces and a piece she thinks is tied for the largest. The only condition is if the piece she had trimmed is still there she had to pick it.

6. Birur chooses next. He chooses a piece he thinks is tied for the largest among the remaining three pieces. If a piece he had trimmed earlier is still unpicked, he has to pick it.

7. It is now Almaz's turn to choose from the remaining two pieces, one that is not trimmed.