I've been thinking about the cake cutting problem recently, with a bit more "physical world" and dexterity in mind. Even in the two-player version, one can add few more criterion.

* Cutting a cake is a skill; it takes an effort (thus associated cost) and you may not be accurate at cutting it. There may be some error bound epsilon.

* If so, in the traditional "I cut you choose" model, as there are risks associated with cutting (what if you intend fairness, but you ended up with lopsided 49:51 cake due to an error?), one may not volunteer to cut the cake; if so, how can we encourage / compensate one to cut the cake?

* The "skill" can go both ways; it takes an effort to figure out which cake is bigger - even more then the cut itself. both parties may volunteer to cut the cake. if so, how can we encourage / compensate one to be the chooser?