Non-Monotonicity in Fair Division of Graphs

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 ≥ 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.