Abstract
We study the online allocation of divisible items to n agents with additive valuations for p-mean welfare maximization, a problem introduced by Barman, Khan, and Maiti~(2022). Our algorithmic and hardness results characterize the optimal competitive ratios for the entire spectrum of -∞ ≤ p ≤ 1. Surprisingly, our improved algorithms for all p ≤ 1/log n are simply the greedy algorithm for the Nash welfare, supplemented with two auxiliary components to ensure all agents have non-zero utilities and to help a small number of agents with low utilities. In this sense, the long arm of Nashian allocation achieves near-optimal competitive ratios not only for Nash welfare but also all the way to egalitarian welfare.