Networked Fairness in Cake Cutting

TL;DR

This paper extends classical cake-cutting fairness concepts to a networked setting, proposing new algorithms for envy-free and proportional allocations based on graph structures, including trees and descendant graphs.

Contribution

It introduces a graphical framework for fair division, generalizes envy-freeness and proportionality to networked agents, and provides new algorithms for specific graph classes.

Findings

Envy-free allocations on trees via a moving-knife algorithm.

Discrete, bounded algorithm for proportionality on descendant graphs.

Simpler envy-free algorithm compared to previous methods.

Abstract

We introduce a graphical framework for fair division in cake cutting, where comparisons between agents are limited by an underlying network structure. We generalize the classical fairness notions of envy-freeness and proportionality to this graphical setting. Given a simple undirected graph G, an allocation is envy-free on G if no agent envies any of her neighbor's share, and is proportional on G if every agent values her own share no less than the average among her neighbors, with respect to her own measure. These generalizations open new research directions in developing simple and efficient algorithms that can produce fair allocations under specific graph structures. On the algorithmic frontier, we first propose a moving-knife algorithm that outputs an envy-free allocation on trees. The algorithm is significantly simpler than the discrete and bounded envy-free algorithm recently designed by Aziz and Mackenzie for complete graphs. Next, we give a discrete and bounded algorithm for computing a proportional allocation on descendant graphs, a class of graphs by taking a rooted tree and connecting all its ancestor-descendant pairs.