If you thought that you knew how to cut a cake, you need to take a serious reality check, for you may have been doing it wrong all these years.
The ‘you cut, I choose’ method - which has been known since time immemorial and has been used ever since to solve children's squabbles over birthday cake, has just been branded as the wrong way of cutting a cake by three authors- Steven J. Brams, Michael A. Jones, and Christian Klamler in an article to be published in the December issue of the ‘Notices of the AMS’.
The ‘you cut, I choose’ method can be explained using the following example - suppose a cake is to be divided between two people, X and Y, a fair procedure would be to have X cut the cake and then have Y choose whichever piece he prefers. Although X would have an incentive to cut the cake exactly in half, she/he will be left with whichever piece Y does not take.
The three authors said that although the ‘you cut, I choose’ method has the desirable property of ‘envy-freeness’ ie neither person envies the other, because each knows he has gotten at least half the cake, the method lacks another desirable property, that of equability.
They explained the lack of equability by using the same example again - suppose one half of the cake is frosted with vanilla icing and the other with chocolate icing, and suppose X values chocolate icing twice as much as vanilla. It is possible that X valuation of the piece she gets will be less than Y’s valuation of his piece, making these the two valuations inequitable.
Brams, Jones, and Klamler described a new method for cake cutting, which they call SP (Surplus Procedure).
Using SP, the cake can be cut in such a way that the value X puts on her piece is approximately the same as the value Y puts on his. They said that by using this method both of them would feel that they are getting about 65% of what they want!
They further said that for cake division among 3 people, there is an extension of SP, called EP (Equability Procedure), that ensures all three get, say, 40% of what they want.
However, it is not always possible to ensure both equability and envy-freeness for divisions among 3 or more people. One desirable property that SP and EP share is that they are ‘strategy-proof,’ ie the players cannot assuredly manipulate these procedures to their advantage.
While explaining SP and EP, Brahn said: "We are proposing a new, more scientific approach to dispute resolution. Even if it is not directly applicable, the reasoning that goes into fair-division algorithms is valuable. It shows how mathematics can contribute to making dispute resolution more rigorous and precise."
indiatimes
The scientific way to cut a cake!
