Dividing a cake

How to divide a cake fairly between two? One way is for one person to divide it and the other to choose. This is widely regarded as being envy-free. For the mathematical idealisation of the problem, this is a theorem. But, as Wikipedia seems to suppose, is what is a theorem in mathematics a reasonable guide to actual cake-cutting? And could some other logic/mathematics help us to decide?

One way to proceed for this all too common type of problem is to consider the axioms of the mathematics logically, not just consider things common-sensibly. In doing so there is always a gap between the mathematics and the application, which hopefully can be filled by science or ‘sound and appropriate judgement’ rather than simply on ‘authority’ or some specious argument.

The first step is typically to look at the axioms and conclusions and identify possible areas where there could be errors, and then to look for examples. Here I have one I prepared earlier:

If a method (e.g. ‘divide and choose’) is logically ‘envy-free’ and is accepted as such by both parties and implemented properly, then by Wikipedia argument would seem to suggest that the results ought to be accepted by both parties, without envy. In my experience this is too often not the case. Presumably, then, the envious party (perhaps me) is being unreasonable? You might like to think about this, treating it as a puzzle: if someone is envious after the use of a method which all agreed to be envy-free, are they in some way ‘bad’?

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Alice has baked a round cake. It is uniform and round apart from 3 walnuts on the top and 3 cherries hidden in it. Bob and Carol have identical preferences, but Carol doesn’t know about the cherries. Bob knows that he can easily cut the cake in half in an objectively fair way, but which Carol would regard as unfair, with one side having more walnuts. Carol thinks that the fair way is for one side to have two walnuts but less cake. Bob knows this.

Bob divides the cake so that the side with two walnuts has more cake than Carol thinks is fair but less than would be objectively fair. Carol chooses the side that she prefers. Neither Bob nor Carol is envious of the other, until perhaps Carol discovers that Bob has all the cherries, in which case she is envious. Is she unreasonable? If not, how has our mathematics ‘failed’? Again, you might like to think about it … .

My take on it is that the mathematics is irrelevant if either party gains some additional information, since then what seemed fair may seem unfair, leading to reasonable envy. How, then to do divide?

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A possible procedure is for one party to divide the cake, the other to choose at random. In the long run, even if the first party has additional information, this is fair to the second party. If the first party has better information than the second, their best strategy – knowing how the cake is to be chosen/allocated, is to divide fairly. If both parties agree that the first party has the best information, then possibly both parties would agree to them dividing the cake. Otherwise the ‘fisrt party’ could be chosen at random somehow. If the allocation is done by a third party, randomly, this is also fair to both parties in the long run, as long as both parties can somehow ‘know’ that the allocation genuinely was random. Of course, if the outcome has favoured one party in the short term, the other party might still actually be envious, but this ought not to happen in the long run.

Optionally, after the choice/allocation, both parties could inspect both sides and then consider swapping. This would favour a party who is better at inspecting or at understanding what is in their own interest, and so may not eliminate envy. But both sides might like to do this. (It makes more sense when there are more than two parties).

This approach can be generalized, but this raises further issues.

 

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