Greedy-Based Online Fair Allocation with Adversarial Input: Enabling Best-of-Many-Worlds Guarantees

TL;DR

This paper demonstrates that the PACE algorithm, originally designed for stochastic inputs, also performs robustly under adversarial conditions, providing strong fairness and efficiency guarantees in online allocations.

Contribution

It establishes that PACE and integral greedy algorithms achieve bounded envy and Nash welfare ratios under certain adversarial models, extending their guarantees to a broader input spectrum.

Findings

PACE is equivalent to the integral greedy algorithm in equal-budgets case.

Both algorithms have bounded envy and Nash welfare ratios under restricted adversarial models.

The results provide a unified guarantee across stochastic and adversarial input models.

Abstract

We study an online allocation problem with sequentially arriving items and adversarially chosen agent values, with the goal of balancing fairness and efficiency. Our goal is to study the performance of algorithms that achieve strong guarantees under other input models such as stochastic inputs, in order to achieve robust guarantees against a variety of inputs. To that end, we study the PACE (Pacing According to Current Estimated utility) algorithm, an existing algorithm designed for stochastic input. We show that in the equal-budgets case, PACE is equivalent to the integral greedy algorithm. We go on to show that with natural restrictions on the adversarial input model, both integral greedy allocation and PACE have asymptotically bounded multiplicative envy as well as competitive ratio for Nash welfare, with the multiplicative factors either constant or with optimal order dependence on the number of agents. This completes a "best-of-many-worlds" guarantee for PACE, since past work showed that PACE achieves guarantees for stationary and stochastic-but-non-stationary input models.