5.3: How not to divide with 3 parties
When first approaching the question of 3-party fair division, it is very tempting to propose this method: Randomly designate one participant to be the divider, and designate the rest choosers. Proceed as follows:
1) Have the divider divide the item into 3 pieces
2) Have the first chooser select any of the three pieces they feel is worth a fair share
3) Have the second chooser select either of the remaining pieces
4) The divider gets the piece left.
Suppose we have three people splitting a cake. We can immediately see that the divider will receive a fair share as long as they cut the cake fairly at the beginning. The first chooser certainly will also receive a fair share. What about the second chooser? Suppose each person values the three pieces like this:
\(\begin{array}{|l|l|l|l|}
\hline & \textbf { Piece 1 } & \textbf { Piece 2 } & \textbf { Piece 3 } \\
\hline \textbf { Chooser 1 } & 40 \% & 30 \% & 30 \% \\
\hline \textbf { Chooser 2 } & 45 \% & 30 \% & 25 \% \\
\hline \textbf { Divider } & 33.3 \% & 33.3 \% & 33.3 \% \\
\hline
\end{array}\)
Solution
Since the first chooser will clearly select Piece 1, the second chooser is left to select between Piece 2 and Piece 3, neither of which she values as a fair share (1/3 or about 33.3%). This example shows that this method does not guarantee a fair division.
To handle division with 3 or more parties, we’ll have to take a more clever approach.