Which Unbounded Protocol for Envy Free Cake Cutting is Better?

Abstract

A division of a cake by n people is envy free if everyone thinks they got the biggest pieces. Note that peoples tastes can differ. There is a discrete protocol for envy free division for n=3 which takes at most 5 cuts. For n=4 and beyond there is a protocol but the number of cuts it takes is unbounded. In particular the number of cuts depends on peoples tastes. Given any number N peoples tastes can be set so that the algorithm takes over N cuts. There are three such algorithms known. Which is better? We have devised a way to measure the number of cuts even though it is unbounded. We use ordinals; therefore, a statement like "this protocol takes at most 2omega steps" makes sense. We analyse all three discrete algorithms for envy free cake cutting with this measure.