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

2025-04-18

AI Summary

This paper studies online allocation of divisible goods to $n$ agents with additive utilities, aiming to maximize the $p$-mean social welfare for $p in [-infty,1]$. We propose a Nash-welfare-based greedy framework that unifies optimization across the entire $p$-spectrum via two key mechanisms: a zero-utility guarantee and local utility rebalancing. Our main contribution is the first demonstration that a lightweight extension of the Nash allocation achieves near-optimal competitive ratio for all $p leq 1/log n$, establishing its universal dominance—from Nash welfare ($p=0$) to egalitarian (min-max) welfare ($p o -infty$). We fully characterize the exact optimal competitive ratio for every $p$, and in the regime $p leq 1/log n$, our algorithm attains asymptotically tight guarantees, significantly improving upon prior algorithms tailored to specific $p$-values.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Machine Learning: Online Learning & Bandits

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Online allocation of divisible items to agents
• Maximizing p-mean welfare for -∞ ≤ p ≤ 1
• Optimal competitive ratios using Nashian allocation

Innovation

• Uses greedy algorithm for Nash welfare
• Adds auxiliary components for non-zero utilities
• Optimizes competitive ratios for p-mean welfare