IIRC the protocol for a three-way split is: One party gradually moves the knife, pointing downward and poised to slice, over the surface of the cake, until either of the other two says "cut". The person who calls "cut" gets that slice. The remainder of the cake is divided according to the cut-and-choose protocol.

Wouldn't the two people not cutting wait 'til the knife was in the middle, and then try to yell 'cut!' as fast as they could? First person to yell "cut" would get half of the cake, and the other two would get a quarter apiece...

All parties can say cut, including the cutter. If he calls it, he gets the part behind the knife, just like any of the others.

In the above scenario, the cutter has an incentive to yell cut as soon as the knife passes the 1/3 point, then each of the remaining people has an incentive to yell cut when it's passed half of the remainder (or 1/3 of the total).

This assumes that the cake is uniform, and most cakes aren't.

If the cutter prefers frosting and knows that the other person doesn't, the cutter can arrange to get a larger share than she'd get if roles were reversed.

I think that the problem can be fixed by giving all participants knives so they're all cutters.

I'm assuming this algorithm isn't: "Give each party a knife. Survivor gets the cake."

That works too.

However, I was trying to say "everyone does the moving knife thing and anyone can call the cut on anyone's knife".

These lecture notes give a decent description of several different ways for three people to share (including non-"moving knife" solutions):

http://www.cs.cmu.edu/afs/cs/academic/class/15451-f05/www/le...