Attempting to share out a cake can easily provoke feelings of injustice. But now a team of mathematicians claims to have found a perfectly fair cake-cutting procedure1.  

Their method, they say, incites no envy among the recipients. In principle, it could be used to divide up more than just cakes, including goods or land in legal disputes. When doling out food, land or money, it can be tough for someone to ensure both that everyone gets a fair deal and that they themselves perceive the deal to be fair. This is especially true when the various parties place different values on different parts of the goods being divvied up.

 Political scientist Steven Brams of New York University and his mathematician and economist colleagues say that apportioning is 'perfect' only if it is efficient, equitable and envy-free. 'Efficient' means that the allocation cannot be made better for any one party while remaining at least as good for all the others.

'Equitable' means that every party values the portion it receives as much as every other party values theirs. And 'envy-free' means that each party thinks it receives the best part, or at least one of several equally good portions.

A common way to divide a cake between two people is that one person cuts, and the other chooses which of the two portions to take. But this isn't always a perfect solution. For example, imagine a square cake where the left half is covered in chocolate and right half is plain. If person A values the chocolate half of the cake more highly than the other, while person B values all parts equally, then either A or B will think they have been given a good or a bad deal regardless of who makes a vertical cut in the cake.

Piece of cake In Brams and colleagues' procedure, the two parties tell a referee how much they covet each part of the cake, and then the referee makes two vertical cuts - each splitting the cake's value exactly in half according to the preferences of each party. If the two cuts are identical, the problem is solved: each party gets an acceptable piece. In general, however, the cuts are different. Then, either one person gets the remaining piece, and compensates the other financially according to the other's valuation of it, or the referee calculates how to cut up the remainder and divides it to the satisfaction of both parties. Brams and colleagues show that this procedure also enforces honesty.

It isn't intuitively obvious, but the team calculates that if either party lies about how much it values different parts of the cake, hoping to trick the referee into giving them more than their fair share, they will in fact end up worse off.

This cake-cutting method can only be made 'perfect' for two or three parties; the researchers have not been able to find such a solution for four. Most disputes over goods or property involve only two or three parties, they point out. Although for cake, they admit, it may be another matter.

References:

Brams, S. J., Jones, M. A. & Klamler, C. Perfect cake-cutting procedures with money. Preprint, submitted to American Mathematical Monthly, (2003).