Approximate Proportionality in Online Fair Division

TL;DR

This paper investigates the online fair division problem, focusing on the approximation of proportionality up to one good (PROP1), and introduces algorithms and impossibility results under various adversarial models and information assumptions.

Contribution

It demonstrates the limitations of greedy algorithms for PROP1, explores the effectiveness of random allocation and prediction-based algorithms, and establishes inapproximability results for stronger fairness notions.

Findings

Greedy algorithms fail to guarantee PROP1 in adaptive adversarial settings.

Random allocation achieves meaningful PROP1 approximation with high probability.

Prediction-based algorithms can robustly approximate PROP1 under certain conditions.

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.