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

2025-11-05

AI Summary

This paper studies fair allocation of indivisible items—comprising both goods and chores—among multiple groups of agents, where agents within each group share identical additive valuations. It focuses on the existence threshold: whether a finite upper bound μ exists such that an envy-free allocation is guaranteed whenever the quantity of each item type is at least μ. The authors establish, for the first time, a computable explicit upper bound μ applicable to any number of groups and item types. Their approach combines combinatorial optimization with non-constructive existence arguments, achieving both conceptual simplicity and partial constructiveness. Crucially, the result unifies treatment of goods and chores and naturally extends to continuous fair division settings (e.g., cake-cutting). This advances the theoretical foundations of multi-group fair division and provides a precise feasibility boundary for algorithm design.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $mu$ such that, if each item type appears at least $mu$ times, an envy-free allocation exists. Their proof is non-constructive and only provides explicit upper bounds on $mu$ 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 $mu$ 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.

Problem

• Deriving explicit bounds for envy-free allocations with multiple groups
• Extending fair division techniques to handle indivisible goods and chores
• Providing constructive solutions for fair allocation with dissimilar preferences

Innovation

• Explicit bounds for arbitrary groups and types
• Simpler technique for constructive guarantees
• Extension to chores and continuous domains