Tight Asymptotic Bounds for Fair Division With Externalities

2026-01-19

AI Summary

This study addresses fair allocation of indivisible goods in the presence of externalities, where agents derive positive or negative utility from others’ allocations, rendering traditional envy-freeness (EF) generally unattainable. The work establishes the first tight asymptotic bounds for approximate fairness in such settings: it proves that for any $n$ agents, an EF-$O(\sqrt{n})$ allocation always exists and can be constructed in polynomial time, and provides a matching $\Omega(\sqrt{n})$ lower bound under binary valuations, demonstrating tightness. Furthermore, it shows that EF1 allocations do not universally exist in this context. Combining techniques from combinatorial optimization, game-theoretic utility modeling, and asymptotic analysis, the paper lays a theoretical foundation and delivers efficient algorithms for fair division with externalities.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We study the problem of allocating a set of indivisible items among agents whose preferences include externalities. Unlike the standard fair division model, agents may derive positive or negative utility not only from items allocated directly to them, but also from items allocated to other agents. Since exact envy-freeness cannot be guaranteed, prior work has focused on its relaxations. However, two central questions remained open: does there always exist an allocation that is envy-free up to one item (EF1), and if not, what is the optimal relaxation EF-$k$ that can always be attained? We settle both questions by deriving tight asymptotic bounds on the number of items sufficient to eliminate envy. We show that for any instance with $n$ agents, an allocation that is envy-free up to $O(\sqrt{n})$ items always exists and can be found in polynomial time, and we prove a matching $\Omega(\sqrt{n})$ lower bound showing that this result is tight even for binary valuations, which rules out the existence of EF1 allocations when agents have externalities.

Problem

• fair division
• externalities
• envy-freeness
• indivisible items
• EF1

Innovation

• fair division
• externalities
• envy-freeness
• asymptotic bounds
• indivisible items