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.
- 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.
- 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.
- 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
- 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.
- 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
- 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.
- It is now Almaz's turn to choose from the remaining two
pieces, one that is not trimmed.
Who Envies Whose Piece?
What do we do with the Remaining piece and the trimmings?