The Long Arm of Nashian Allocation in Online $p$-Mean Welfare Maximization

TL;DR

This work investigates online allocation of divisible items to $n$ agents with additive valuations, aiming to maximize the $p$-mean welfare for the full range $-\,\infty \le p \le 1$ under unit monopolist utilities. It introduces Nashian allocation-based algorithms that, remarkably, yield near-optimal competitive ratios across multiple $p$-regimes: $1/p$-competitiveness for large $p$, $O(\log n)$ in the Nashian window $-o(1) \le p \le 1/\log n$, and polynomial bounds for negative $p$, with a bridging framework from Nashian to harmonic to egalitarian welfare. The core technique is the Fundamental Lemma of Nashian Allocation, together with a two-copy relaxation and auxiliary components (uniform base utilities and regularized egalitarian steps) that unify the analysis across regimes. The paper also provides hardness results that nearly match the algorithmic bounds, showing tight characterizations in the positive and negative regimes, and extends the results to non-unit monopolist utilities. Collectively, the results reveal that Nash welfare maximization serves as a robust, unifying heuristic underpinning online p-mean welfare maximization across a broad spectrum of fairness-efficiency tradeoffs, with practical implications for fair and efficient online allocation.

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 $-\infty \le p \le 1$. Surprisingly, our improved algorithms for all $p \le \frac{1}{\log n}$ 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.

Figures (2)

• Figure 1: Illustration of the hard instance for Nashian and positive regimes, i.e., when $p \ge -1/\log n$.
• Figure 2: Illustration of the family of hard instances for the negative regime. Here, $L \ge 1$ is the number of groups of good agents. A decreasing sequence $1=s_0>s_1>\dots>s_L\geq 0$ determines the group sizes. Rounded rectangles represent groups of agents, labeled by the groups' indices and sizes. Circles represent items, labeled by the items' supplies. Having an edge between a group of agents and an item means that the agents have value $1$ for the item; the parallel edges between the good agents and the corresponding items in the Makeup Stage indicate having a separate item for every such agent. The items in the Upper Triangular Stage arrive first from left to right; then, those in the Makeup Stage arrive by an arbitrary order. The groups of agents and items are colored according to the offline optimal allocation. For example, the items in the Upper Triangular Stage are colored red because the offline benchmark allocates them to group $B$ which is also colored red.

Theorems & Definitions (65)

• lemma 1
• lemma 2
• theorem 3
• corollary 4
• lemma 5: Fundamental Lemma of Nashian Allocation
• proof
• proof : Proof of \ref{['thm:nashian']}
• theorem 6
• definition 1: Bad Agents
• lemma 7