Fair Division 1

What I've been considering is the problem where each person knows exactly how much everybody else values every bit of the cake to be divided - and they are all very intelligent and greedy and will break a promise if it suits them. The usual scenario for game theory problems in fact where they lose out badly in the prisoners' dilemma.

This is quite different from the usual cake fair division problem. In the normal problem with two people dividing a cake equally using cut and choose the one choosing has the advantage, but in the full information case the cutter has a large advantage.

What I have devised is schemes whereby in the two and three person cases the cake can be divided among the people so they each get the maximum possible common fraction of the value they put on the cake. I'm not absolutely certain yet the three person case is proof against cheating but I'm fairly confident. The only real restriction is that all the cake must be worth something to everyone.

For the two player case the solution is quite straightforward

The algorithm can be extended quite easily to deal with where one person is entitled to getting for example twice as much by his valuation as the other person did by theirs.

The three person case is considerably more complicated but I can summarize it as:

The worry about cheating is whether anyone can skew the offers much at any stage for example whether B could offer less to C on the basis that it would be in C's interests not to show there is a problem with the fraction W. I've gone through the various stages and I can't see any possibility but I have had a number of schemes I investigated before which did prove susceptible.