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Tight Asymptotic Bounds for Fair Division With Externalities

Frank Connor, Max Dupré la Tour, Vishnu V. Narayan, Šimon Schierreich

TL;DR

The paper addresses fair division with externalities, where agents’ utilities depend on others’ allocations, and EF1 may fail. It introduces an equivalence to an asymmetric envy model and a multicolor discrepancy framework to derive a polynomial-time, tight $O( obreak \, ext{sqrt}(n))$ upper bound on the envy-relaxation EF-$k$, and proves a matching lower bound $ obreak \Omega( ext{sqrt}(n))$ even for binary/no-chores valuations, thereby resolving open questions about the existence of EF1 allocations. Additionally, it provides an ex-ante truthful consensus mechanism achieving EF-$ obreak O(n)$. These results settle key conjectures and offer a principled approach for handling externalities in fair division with indivisible items.

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 $Ω(\sqrt{n})$ lower bound showing that this result is tight even for binary valuations, which rules out the existence of EF1 allocations when agents have externalities.

Tight Asymptotic Bounds for Fair Division With Externalities

TL;DR

The paper addresses fair division with externalities, where agents’ utilities depend on others’ allocations, and EF1 may fail. It introduces an equivalence to an asymmetric envy model and a multicolor discrepancy framework to derive a polynomial-time, tight upper bound on the envy-relaxation EF-, and proves a matching lower bound even for binary/no-chores valuations, thereby resolving open questions about the existence of EF1 allocations. Additionally, it provides an ex-ante truthful consensus mechanism achieving EF-. These results settle key conjectures and offer a principled approach for handling externalities in fair division with indivisible items.

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- 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 agents, an allocation that is envy-free up to items always exists and can be found in polynomial time, and we prove a matching lower bound showing that this result is tight even for binary valuations, which rules out the existence of EF1 allocations when agents have externalities.
Paper Structure (13 sections, 13 theorems, 74 equations)

This paper contains 13 sections, 13 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. Preliminaries
  5. Fairness in the Externalities Model
  6. Technical Tools
  7. Asymmetric Envy Model
  8. Fairness in the Asymmetric Envy Model
  9. Multicolor Discrepancy and Fair Division
  10. An ${\mathcal{O}\left(\sqrt{n}\right)}$ Upper Bound
  11. An Asymptotically Tight Lower Bound
  12. An Ex-Ante Truthful Mechanism
  13. Conclusions

Key Result

Proposition 1

There exists a function $f$ that maps each instance $I$ of the externalities model to an instance $f(I)$ of the asymmetric envy model such that, for every allocation $A \in \mathcal{A}$, the allocation $A$ is EF (respectively, EF-$c$) in $I$ if and only if it is EF$_{a}$ (respectively, EF$_a$-$c$) i

Theorems & Definitions (23)

  • Proposition 1
  • proof
  • Theorem 1: ManurangsiM2026
  • Lemma 1
  • proof
  • Theorem 2: ManurangsiM2026
  • Theorem 3
  • proof
  • Corollary 1
  • proof
  • ...and 13 more