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.

翻译：我们研究一个在线分配问题，其中物品顺序到达，智能体价值由对手选择，目标是在公平性与效率之间取得平衡。我们旨在分析那些在其他输入模型（如随机输入）下具有强保证的算法的性能，从而实现对各类输入的鲁棒性保证。为此，我们研究了为随机输入设计的现有算法PACE（基于当前估计效用的步调调整）。我们证明，在预算相等的情况下，PACE等价于积分贪心分配算法。进一步，我们表明在对对抗性输入模型的自然限制下，积分贪心算法与PACE均具有渐近有界的乘性嫉妒以及针对纳什福利的竞争比，其中乘性因子或为常数，或其关于智能体数量的阶数达到最优。这为PACE补充了“多世界最优”保证——此前的研究已表明PACE在平稳及随机非平稳输入模型下具有保证。