Fair Division via Social Comparison
This addresses fair division in networked settings where agents have limited visibility, which is relevant for resource allocation in social or distributed systems, but it is incremental as it builds on classical cake cutting models.
The paper tackles the cake cutting problem by introducing a variant where agents compare their shares only to neighbors in a social network, defining local envy-freeness and proportionality, and shows that global proportionality does not imply local proportionality. It characterizes graphs for which a single-cutter protocol achieves locally envy-free allocations with O(n^2) query complexity and proves a lower bound of Ω(√n) on the price of envy-freeness for any connected graph.
In the classical cake cutting problem, a resource must be divided among agents with different utilities so that each agent believes they have received a fair share of the resource relative to the other agents. We introduce a variant of the problem in which we model an underlying social network on the agents with a graph, and agents only evaluate their shares relative to their neighbors' in the network. This formulation captures many situations in which it is unrealistic to assume a global view, and also exposes interesting phenomena in the original problem. Specifically, we say an allocation is locally envy-free if no agent envies a neighbor's allocation and locally proportional if each agent values her own allocation as much as the average value of her neighbor's allocations, with the former implying the latter. While global envy-freeness implies local envy-freeness, global proportionality does not imply local proportionality, or vice versa. A general result is that for any two distinct graphs on the same set of nodes and an allocation, there exists a set of valuation functions such that the allocation is locally proportional on one but not the other. We fully characterize the set of graphs for which an oblivious single-cutter protocol-- a protocol that uses a single agent to cut the cake into pieces --admits a bounded protocol with $O(n^2)$ query complexity for locally envy-free allocations in the Robertson-Webb model. We also consider the price of envy-freeness, which compares the total utility of an optimal allocation to the best utility of an allocation that is envy-free. We show that a lower bound of $Ω(\sqrt{n})$ on the price of envy-freeness for global allocations in fact holds for local envy-freeness in any connected undirected graph. Thus, sparse graphs surprisingly do not provide more flexibility with respect to the quality of envy-free allocations.