Non-Monotonicity in Fair Division of Graphs

2025-11-05

AI Summary

This paper studies fair allocation of graph vertices among multiple agents, where each agent’s valuation of a vertex subset (bundle) is defined as its cut value—the number of edges with exactly one endpoint in the subset—yielding a canonical non-monotonic utility function (marginal utilities may be positive, negative, or zero). Fairness is formalized via EF1 (envy-freeness up to one good), and efficiency is captured by transfer stability (TS). The main contribution is a characterization of the compatibility between EF1 and TS: existence of EF1+TS allocations is non-monotonic in the number of agents—guaranteed for two agents, impossible in some instances with three agents, yet restored for four or more. This pattern is established for general graphs. Furthermore, the paper ensures existence and constructibility of EF1+TS allocations for any number of agents either by relaxing efficiency requirements or by restricting the underlying graph to be a forest.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

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 $ngeq 4$. 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.

Problem

• Studying fair division of graph vertices with non-monotonic valuations
• Analyzing existence of EF1 allocations with transfer stability across agent counts
• Providing algorithms for fair allocations in forests and weakened efficiency

Innovation

• Fair vertex allocation using graph cut values
• Envy-freeness with transfer stability guarantees
• Efficient algorithms for forest graphs and weakened efficiency