A cake for three that nobody can call unfair

Question

Three friends share a cake. They have different tastes: one loves the icing, another the fruit, so they value the pieces differently and there is no referee. For two people, "one cuts, the other chooses" guarantees each a half by their own valuation.

Design a procedure for three people so that every person can guarantee themselves at least one third of the cake, by their own valuation, no matter what the others do. Then explain why this does not guarantee that nobody envies anyone else's share.

Accepted Answer

The last-diminisher procedure. 1. Person A cuts a slice that A values at exactly one third of the cake. 2. The slice is offered to B. If B thinks it is worth more than a third, B trims it down to what B considers exactly a third (the trimmings go back to the main cake). If B thinks it is a third or less, B passes it on untouched. 3. The slice, possibly trimmed, is offered to C, who may likewise trim it to what C considers a third, or pass. 4. The slice goes to the last person who trimmed it (or to A if nobody did). That person leaves with it. 5. The remaining two people divide the rest of the…

A cake for three that nobody can call unfair

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