Networked Fairness in Cake Cutting

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 in [Aziz and Mackenzie, 2016a] 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.

Cake Cutting on Graphs: A Discrete and Bounded Proportional Protocol

“…This section briefly discusses locally envy-free allocations on trees. Previously, a continuous locally envy-free protocol on trees was already known [8]. However, despite of the globally envy-free protocol designed in [5], a simple discrete and bounded one on trees is still unknown.…”

“…For local envy-freeness, a nontrivial continuous protocol for trees was proposed in [8] when the protocol is allowed to perform the so-called Austin Cut Procedure [3]. However, this protocol is hard to generalize.…”

“…In the meanwhile, for any graph, the existence of a locally proportional allocation is guaranteed from that of a globally envy-free allocation. In [8], the authors gave a simple locally proportional protocol for a special class of graphs known as the descendant graph. It is listed both in [1] and [8] as an open question to design simple and locally proportional protocols for more classes of graphs.…”

“…The notion of local fairness with respect to social networks was first introduced to the cake cutting problem independently in [1] and [8]. Previously known algorithmic results are mainly from [8], which gave a locally envy-free continuous protocol for trees as well as a locally proportional protocol for a special class of graphs known as the descendant graph.…”

“…Local proportionality generalizes proportionality and is in an interesting middle ground between proportionality and envy-freeness: its existence is guaranteed by that of an envy-free allocation, but no simple protocol is known to produce such a locally proportional allocation for general graphs. Previous works showed locally proportional protocols for special classes of graphs, and it is listed in both [1] and [8] as an open question to design simple locally proportional protocols for more general classes of graphs. In this paper we completely resolved this open question by presenting a discrete and bounded locally proportional protocol for any given graph.…”