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On the Existence of Fair Allocations for Goods and Chores under Dissimilar Preferences

Egor Gagushin, Marios Mertzanidis, Alexandros Psomas

TL;DR

The paper resolves a major open question by providing explicit upper bounds on the copy threshold $\mu$ guaranteeing envy-free allocations for arbitrary numbers of item types $t$ and groups $d$, under a dissimilarity condition between group valuations. It introduces the Relative Norm mechanism, a rounding framework, and a Brauer coin-problem-based technique that yield constructive envy-free allocations for goods, chores, and cake cutting, with extensions to stochastic settings. The results generalize prior existential bounds and recover state-of-the-art proportionality guarantees up to constants, while connecting fairness existence to measurable dissimilarity (via $f$-divergences) between groups. The approach is broad and algorithmic, enabling practical constructions and offering a unifying lens for fair division across multiple models, including trading-post and inverse trading-post mechanisms. Overall, the work advances both theory and potential practice in fair allocation of indivisible resources across diverse domains.

Abstract

We study the fundamental problem of fairly allocating a multiset $\mathcal{M}$ of $t$ types of indivisible items among $d$ groups of agents, where all agents within a group have identical additive valuations. Gorantla et al. [GMV23] showed that for every such instance, there exists a finite number $μ$ such that, if each item type appears at least $μ$ times, an envy-free allocation exists. Their proof is non-constructive and only provides explicit upper bounds on $μ$ for the cases of two groups ($d=2$) or two item types ($t=2$). In this work, we resolve one of the main open questions posed by Gorantla et al. [GMV23] by deriving explicit upper bounds on $μ$ that hold for arbitrary numbers of groups and item types. We introduce a significantly simpler, yet powerful technique that not only yields constructive guarantees for indivisible goods but also extends naturally to chores and continuous domains, leading to new results in related fair division settings such as cake cutting.

On the Existence of Fair Allocations for Goods and Chores under Dissimilar Preferences

TL;DR

The paper resolves a major open question by providing explicit upper bounds on the copy threshold guaranteeing envy-free allocations for arbitrary numbers of item types and groups , under a dissimilarity condition between group valuations. It introduces the Relative Norm mechanism, a rounding framework, and a Brauer coin-problem-based technique that yield constructive envy-free allocations for goods, chores, and cake cutting, with extensions to stochastic settings. The results generalize prior existential bounds and recover state-of-the-art proportionality guarantees up to constants, while connecting fairness existence to measurable dissimilarity (via -divergences) between groups. The approach is broad and algorithmic, enabling practical constructions and offering a unifying lens for fair division across multiple models, including trading-post and inverse trading-post mechanisms. Overall, the work advances both theory and potential practice in fair allocation of indivisible resources across diverse domains.

Abstract

We study the fundamental problem of fairly allocating a multiset of types of indivisible items among groups of agents, where all agents within a group have identical additive valuations. Gorantla et al. [GMV23] showed that for every such instance, there exists a finite number such that, if each item type appears at least times, an envy-free allocation exists. Their proof is non-constructive and only provides explicit upper bounds on for the cases of two groups () or two item types (). In this work, we resolve one of the main open questions posed by Gorantla et al. [GMV23] by deriving explicit upper bounds on that hold for arbitrary numbers of groups and item types. We introduce a significantly simpler, yet powerful technique that not only yields constructive guarantees for indivisible goods but also extends naturally to chores and continuous domains, leading to new results in related fair division settings such as cake cutting.

Paper Structure

This paper contains 27 sections, 35 theorems, 41 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Chores and envy-freeness.
  4. Cake Cutting.
  5. Proportionality for Goods.
  6. Future Directions.
  7. Related Work
  8. Relaxations of Proportionality and Envy-Freeness.
  9. Fair allocations in stochastic settings.
  10. Preliminaries
  11. Allocations.
  12. Fairness Notions
  13. Statistical Distances
  14. Conditions Under Which Envy-Free Allocations Exist
  15. Extensions
  16. ...and 12 more sections

Key Result

Theorem 1

Let $n_1 \le n_2 \le \cdots \le n_d$, be positive integers such that $\gcd(n_1, \dots, n_d) = 1$. Then, any integer $k \ge (n_1-1)(n_d-1)$ can be written as $k = \sum_{i =1}^d x_i \cdot n_i$, where $x_i\ge0$ is an integer for all $i \in [d]$.

Theorems & Definitions (65)

  • Theorem 1: brauer1942Partitions
  • Corollary 1
  • Definition 1: gcd $g$ and threshold $\theta$
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 2
  • ...and 55 more