To ensure fair distribution of portions of cake (or whatever) between two people, there is a method called "cut and choose". One person cuts the cake, the other person chooses which of the two pieces goes to whom. It's a way of ensuring fairness without the need for a third party or authority figure. The problem is that it only applies to situations involving two people, and quite often there are three or more people involved. I've wondered for a while whether it is possible to extend the method to deal with an arbitrary number of people.
One way might be to have one person cut the first piece of cake, then someone else decides whether to take that piece or to have the cutter take it. Once you take your piece, you're out of the process (to enjoy your cake) and the other person in your cut-and-choose pair continues as cutter for the next piece. The problem is that it's really tricky, for instance, to judge a fair 1/7th of a cake by eye, so the first few cuts are likely to be very inaccurate. It's still as fair as you can get, though, because you either cut a fair piece or you lose. Cut too big and that much cake is gone from the game. Cut too small and you leave with less. The problem would be if 1/6th of the cake looked close enough to be fair, which has a carry-on effect to the rest of the cutting.
I'd love to give this experiment a try with real people, just to see if it works.
Mokalus of Borg
PS - I'm sure "share some cake with me" will be a pretty easy sell for experimental subjects.
PPS - However, "please carefully follow this procedure before eating" might be more difficult.