This protocol for fair cake division works in a wide variety of situations. Since there are only 2 players, no coalitions are possible, which of course makes things simpler.

Here's the protocol: Alice cuts the cake into 2 pieces (which can be any size!); then Bob picks his piece. Note that Bob's piece will never be smaller than Alice's (otherwise he'd have picked the other piece). Thus Bob is convinced the division was "fair". On the other hand, Alice can always divide the cake equally, thus getting as much as Bob; anything else leads her to get less cake. Thus the cake is split equally, and neither party can complain about cheating!

The point is valid, but I'm afraid it's not likely to make sense in the world of real cakes.

I'm two years older than my brother, so for ten years or so we competed for things like cakes directly rather than through sarcasm. Our parents, thinking themselves fair people, would tell us to use the one-divides-and-the-other-chooses method to split things. Since I was older and thus presumably better at eyeballing portions, I always cut — and discovered that this method was terribly unfair, because in practice I could never cut the cake quite evenly, no matter how much I trimmed from one part and moved to the other.

And thus, in the real world, Alice can't always divide equally. There will be some error large enough to notice, and Bob will get the bigger part. It would work (over the long run) if Alice and Bob switched jobs each time there was something to be halved. They could even switch back and forth on each cake, by alternately making hypothetical cuts in a sort of double-auction.

Actually, there's a word for switching back and forth like that: bargaining. Alice indicates a possible cake division and chooses one of the parts for herself; Bob refuses and offers her a different (and presumably slightly more even) portion, et cetera.

Very few resource disputes are over a homogenous mass of anything; if they were, it would be trivial to make perfect halves. More often, people argue over something with several kinds of inherent goodies (both icing and filling, say, or a house and a car and a table-setting), and therefore many distinct pairs of 'halves'. It's likely that Alice and Bob want different things out of their slices, so they'd be better off discussing (however obliquely and guardedly) exchanges than agreeing to a compromise without feedback.

In other words, division is more like barter than like monetary sale. If you expect someone to buy your foo with some form of currency, you might put a fair-looking price on it and let them take it or leave it. However, if your customers are pre- or post-industrial, you'll want to inspect the poultry and CDRs or other goods that they offer in trade before you suggest a deal. For instance, you can give someone foo for cheap if they give you raw materials for cheap. I can give my brother more icing if he agrees to let me take more filling.

No iterative method with positive feedback will be worse than flipping a coin to pick the cutter and again to pick the picker, which is what the above writeup amounts to.

In yet more words: in any negotiation, you should have a clear idea of what the other person has and wants before you make a final offer. Bargaining is usually worth your time.

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