Which Unbounded Protocol for Envy Free Cake Cutting is Better?

TL;DR

This paper compares three envy-free cake-cutting algorithms for multiple people using ordinal measures to evaluate their unbounded number of cuts, providing a new way to analyze their efficiency.

Contribution

It introduces a novel ordinal-based measurement method to compare unbounded cake-cutting protocols and analyzes three existing algorithms under this framework.

Findings

All three algorithms have different ordinal bounds on cuts.

The analysis reveals which algorithm performs better under the ordinal measure.

Provides insights into the complexity of envy-free division for larger groups.

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.