The perfect way to cut a cake

Solved: the perfect way to cut a cake


The art of cake-cutting requires great care and skill to ensure no party is left feeling cheated or envious. Now, however, parents and party hosts can approach the task with a little more confidence – mathematicians claim to have found the perfect way to cut a cake and keep everyone happy.

“The problem of fair division is one of the oldest existing problems. The cake is a metaphor for any divisible object where people value different parts differently,” explains Christian Klamler, at the University of Graz, Austria, who solved the problem with fellow mathematicians Steven Brams and Michael Jones.

According to Klamler, for any division to be acceptable, it must ideally be equal among all parties, envy-free so that no one prefers another’s share and equitable, where each places the same subjective value on their share.

Traditional methods, such as the "you cut, I choose" method, where one person halves the cake and the other chooses a piece, are flawed because though both get equal shares and neither is envious, the division is not equitable - one piece may have more icing or fruit on it than another, for example.
Impartial cutter

Enter the “Surplus Procedure” (SP) for cake-sharing between two people, and the "Equitability Procedure" (EP) for sharing between three or more. Both involve asking guests to tell the cake-cutter how they value different parts of the cake. For example, one guest may prefer chocolate, another may prefer marzipan.

Under SP, the two parties first receive just half of the cake portion that they subjectively valued the most. Then the "surplus" left over is divided proportionally according to the value they gave it. EP works in a similar way: the guests first get an equal proportion of the part of the cake they each value the highest – a third each if they are three; a quarter each if they are four, etc – and then the remainder is again divided along the lines of subjective value.

The result is everyone is left feeling happy, Klamler says. Two people, for example, may feel they are each getting 65% of what they want rather than just half.

“These procedures are new and have never been tried out in real-world applications,” says Brams. “But where there is a divisible good like land or water, which players value differently, the procedure could be used to allocate more-than-proportional shares, making everybody as happy as possible.”

Intriguingly, the procedures are "tamper-proof" – people cannot manipulate the process and must be truthful with the referee, or else they could end up with less than makes them happy.

Journal reference: Notices of the American Mathematical Society (Vol 53, p1014)