Networked Fairness in Cake Cutting
This addresses fairness in resource allocation for networked agents, offering incremental improvements by adapting classical notions to specific graph structures.
The paper tackles fair division in cake cutting by introducing a graphical framework where fairness comparisons are limited to network neighbors, generalizing envy-freeness and proportionality to this setting. It proposes algorithms, including a moving-knife method for trees that is simpler than prior work, and a discrete bounded algorithm for descendant graphs.
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.