Three cooks have each paid one third to purchase a bag of herbs. In the past, two of the cooks have divided their purchases in the following manner: First one cook would divide the herb, by eye, into two piles she considered to be equal. The second cook would then choose the pile she thought was bigger.

By what process may the three cooks divide their herbs in such a way that each was content that she had recieved at least one third of the total? (No scales or other devices are available to assist the division)

We assume that this to be "fair" means that each cook is "reasonable" and that each cook gets at least their fair share. It does not mean that after the division, a cook won't think someone else got more.
With these assumptions, here is one fair division process:
1) Order the cooks 1,2, 3.
2) The first cook seperates out what she considers to be a third of the total pile.
3) The second cook may pass if she feels the third is fair or reduce the portion to what she considers fair.
4) Then the third cook may do the same, pass or reduce.
5) On the second round, the cooks have the choice of take the portion or pass. There is the added rule that the last cook to adjust the portion MUST take the pile; they may not pass.
6) After this point, the remaing herbs can be divided by the remaining two cooks in the manner described in the problem.

Notes: Why is this proces fair? Every cook gets to insure that the first portion meets their personal criteria for "fair share", the portion never increases after their turn at adjusting, and no cook is forced to take a portion if it is adjusted by another cook.
Why is it possible that a cook covet another cooks portion? The cook receiving the first portion may disagree with the division of the final two portions, thus seeing that someone got more than the first portion. This cook should not, however, feel that they received less than a third. Thus this is a fair division and not a maximal division.
Can this be generalizable to any number of cooks? Yes, and it should be fairly straightforward to see how.
Could two cooks get together and cheat? Without violence? No, I don't think so.

Posted by owl on 2006-01-27 11:09:03