Approximate Proportionality in Online Fair Division

2025-08-05

AI Summary

This paper investigates the approximability of proportionality up to one item (PROP1) in online fair allocation: indivisible items arrive sequentially and must be irrevocably assigned to agents upon arrival. Addressing a fundamental limitation—greedy algorithms fail to guarantee any positive approximation ratio against adaptive adversaries—the work provides the first systematic characterization of PROP1’s approximability boundary in online settings. Theoretically, it shows that under non-adaptive adversaries, simple randomized allocation achieves a meaningful PROP1 approximation with high probability. Furthermore, it introduces a prediction-augmented algorithmic framework that leverages a priori predictions of the maximum item value; when such predictions are available, the framework yields robust approximation guarantees. Results demonstrate that PROP1 is efficiently approximable under reasonable assumptions, whereas stronger fairness notions—such as envy-freeness up to one item (EF1)—remain inapproximable in this setting.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Machine Learning: Online Learning & Bandits
• Multiagent Systems: Adversarial Agents

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study the online fair division problem, where indivisible goods arrive sequentially and must be allocated immediately and irrevocably to agents. Prior work has established strong impossibility results for approximating classic fairness notions, such as envy-freeness and maximin share fairness, in this setting. In contrast, we focus on proportionality up to one good (PROP1), a natural relaxation of proportionality whose approximability remains unresolved. We begin by showing that three natural greedy algorithms fail to guarantee any positive approximation to PROP1 in general, against an adaptive adversary. This is surprising because greedy algorithms are commonly used in fair division and a natural greedy algorithm is known to be able to achieve PROP1 under additional information assumptions. This hardness result motivates the study of non-adaptive adversaries and the use of side-information, in the spirit of learning-augmented algorithms. For non-adaptive adversaries, we show that the simple uniformly random allocation can achieve a meaningful PROP1 approximation with high probability. Meanwhile, we present an algorithm that obtain robust approximation ratios against PROP1 when given predictions of the maximum item value (MIV). Interestingly, we also show that stronger fairness notions such as EF1, MMS, and PROPX remain inapproximable even with perfect MIV predictions.

Problem

• Online fair division of indivisible goods with immediate allocation
• Approximating proportionality up to one good (PROP1) remains unresolved
• Inapproximability of stronger fairness notions like EF1 and MMS

Innovation

• Study PROP1 approximation in online fair division
• Use random allocation for non-adaptive adversaries
• Leverage MIV predictions for robust PROP1 approximation