Background
I have (rather recently) dabbled in game theory. I need it to design an algorithm to share chores. Obviously this is a kind of cake-cutting problem. So far, I have fought my way through An Introduction to Game Theory by Martin J. Osborne, but I'm still feel far from comfortable with it. I have a solid foundation in calculus, know how to deal with ODEs and PDEs (but I try to avoid them if I can. :)) And yes, I'm not a mathematician, I"m an engineer.
Problem
The 'cake' needs to be split among $k\geq 2$ players. The twist is that valuation of the players is not finitely additive, it has a maximum i.e. there is an amount of cake that they will find more valuable than a larger amount (kind of being afraid of overeating $-$ or being on a diet).
Question
Does anybody know of a starting point to how tackle this? (Efficient or equitable solutions would be the most interesting.) All the resources I found, treat only the finitely additive case. I would also be grateful for any freely downloadable material.
EDIT: I'm looking for efficient and/or equitable ways of splitting the cake, regardless of the protocol to achieve it. If there also is a protocol for that, the better for me. :)
Best Answer
TLDR: Borgers book is \$35 and awesome for the additive case of compensation. From this you can Lagrange multipliers to solve nice variations. Add "unimodal" to your search to find articles where the marginal utility goes up for a bit and then back down.
I found cake-cutting to have some additional complications that can be solved in a manageable way by simply paying people a little to make sure things are equal. In other words, we have a cake where people have different (usually finitely additive) utility functions and those difference produce the opportunity for extra-good in the division, but we also have a pile of money where everyone's utility function agrees and is plain old "dx", the easiest of all.
Compensation problems with finitely many goods and additive utility functions are very easy to understand, and are worked out in a very nice textbook way in Borger, 2010 (\$35 print, over \$960 online form the publisher, impressive compensation scheme). The techniques used do not change too much with different utility functions if they are not too weird (for instance, unimodal, continuous function in your case, or monotone, continuous in the case I read about), since you are just optimizing differentiable functions of a single variable, and so the maximums occur either at corners or at solutions to Lagrange multiplier constraints. In the additive case, all the utility functions are linear, so the calculus disappears, and the maximums occur at (at least one of the) corners.
I did not find an easily digestible article discussing your case (which is probably very similar to the example you give: adjust compensation to encourage neither under-production nor over-production; "unimodal" should help the search). The cases I read about were "auctions" where the risk involved means utility functions are "sub-linear", so that people are less willing to risk large amounts of money, so the marginal utility of the cash decreases (but never goes negative as I think it does in your case).