Cake Jealousy

When two people want to divide a cake "fairly" among themselves, they have a pretty simple way to do it - one cuts the cake, and the other chooses a piece.

First, think how you'd do it in a three-way, then for n-way. There are a few variations, that you might want to think of:

• Objectivity - given two pieces of the cake, all players would rank them the same - i.e. it is not possible that Alice likes piece A better and Bob likes piece B better.
• Accuracy - all players are able to make fine-tuned cuts.
• Non-cooperativeness - all players want their piece to be the biggest - i.e., two players A, B won't manipulate the protocol so that B gets a huge chunk and A gets a little chunk (even if A+B > 2/n, their respective share), since A does not trust/cooperate with B.

Also, how many cuts would you need for it? Can you do it with the minimal (n-1) cuts?

It's actually a pretty hard question, so I'll give the password for the solution - it's just cake.

Solution