Non-Monotonicity in Fair Division of Graphs

TL;DR

This paper studies fair division of graph vertices with non-monotonic valuations, revealing a complex relationship between the number of agents and the existence of envy-free and transfer-stable allocations, with positive results under certain conditions.

Contribution

It uncovers the non-monotonic relationship between agent count and fair allocation existence, providing algorithms for positive cases and analyzing conditions for fairness.

Findings

EF1 and TS allocations exist for 2 agents

Such allocations may not exist for 3 agents

Existence is guaranteed for all n with relaxed conditions or forest graphs

Abstract

We consider the problem of fairly allocating the vertices of a graph among $n$ agents, where the value of a bundle is determined by its cut value -- the number of edges with exactly one endpoint in the bundle. This model naturally captures applications such as team formation and network partitioning, where valuations are inherently non-monotonic: the marginal values may be positive, negative, or zero depending on the composition of the bundle. We focus on the fairness notion of envy-freeness up to one item (EF1) and explore its compatibility with several efficiency concepts such as Transfer Stability (TS) that prohibits single-item transfers that benefit one agent without making the other worse-off. For general graphs, our results uncover a non-monotonic relationship between the number of agents $n$ and the existence of allocations satisfying EF1 and transfer stability (TS): such allocations always exist for $n=2$, may fail to exist for $n=3$, but exist again for all $n\geq 4$. We further show that existence can be guaranteed for any $n$ by slightly weakening the efficiency requirement or by restricting the graph to forests. All of our positive results are achieved via efficient algorithms.