Non-Monotonicity in Fair Division of Graphs

TL;DR

This paper studies fair division of graph vertices under cut-valuations, a non-monotone bundle-dependent valuation where an item’s marginal contribution can be positive or negative depending on the bundle composition. The authors analyze envy-freeness up to one item (EF1) alongside efficiency notions—Transfer Stability (TS), weak transfer stability (wTS), Social Optimality (SO), and Pareto Optimality (PO)—and reveal a surprising non-monotonic relationship between the number of agents $n$ and the existence of EF1+TS allocations on general graphs: such allocations exist for $n=2$, may fail for $n=3$, and exist again for all $n\ge 4$. They show that EF1 can always be paired with the weaker wTS (polynomial-time) and that a leximin solution achieves $\tfrac{1}{2}$-EF1+PO, while for forests EF1+SO allocations exist and can be computed in polynomial time via a rooting-based algorithm. The results extend to equitable graph cuts and provide structural insights: strong EF1+SO guarantees fail on general graphs but hold in forests; and even modest relaxations (like wTS) restore universal EF1 compatibility, offering practical, efficient procedures for a broad class of instances.

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.

Figures (3)

• Figure 1: Graph with $8$ items (vertices).
• Figure 2: An example showing the incompatibility of EF1 and transfer stability (TS) for $n=3$.
• Figure 3: An illustration of the characterization in \ref{['claim:chore_for_two_agents']}.

Theorems & Definitions (43)

• Example 1: Allocations and cut-valuations
• Remark 1: Relation to Graph Partitions
• Example 2: Understanding Efficiency Notions
• Proposition 1
• proof
• Theorem 1: Non-existence of + allocations
• proof
• Theorem 2: + allocations when $n\ge 4$
• Lemma 1
• Lemma 2