Online Fair Division With Subsidy: When Do Envy-Free Allocations Exist, and at What Cost?

2025-10-15

AI Summary

This paper studies fair allocation of indivisible items in an online setting: items arrive sequentially and must be irrevocably allocated upon arrival, with the objective of achieving envy-freeness (EF) via minimal subsidies. We extend offline subsidy-based EF mechanisms to the online domain for the first time, systematically analyzing feasibility and subsidy cost across additive, k-demand, and SPLC utility models. Theoretically, we establish that EF can be maintained online under additive and multi-category valuations; notably, in several models, the total subsidy remains bounded—i.e., independent of the number of items—and we derive nearly tight upper and lower bounds. Our key contribution lies in uncovering how valuation structure fundamentally governs both the feasibility of online EF and the associated subsidy cost, thereby establishing the first theoretical framework for subsidy-enabled fair allocation in online environments.

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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the problem of fairly allocating $m$ indivisible items arriving online, among $n$ (offline) agents. Although envy-freeness has emerged as the archetypal fairness notion, envy-free (EF) allocations need not exist with indivisible items. To bypass this, a prominent line of research demonstrates that there exist allocations that can be made envy-free by allowing a subsidy. Extensive work in the offline setting has focused on finding such envy-freeable allocations with bounded subsidy. We extend this literature to an online setting where items arrive one at a time and must be immediately and irrevocably allocated. Our contributions are two-fold: 1. Maintaining EF Online: We show that envy-freeability cannot always be preserved online when the valuations are submodular or supermodular, even with binary marginals. In contrast, we design online algorithms that maintain envy-freeability at every step for the class of additive valuations, and for its superclasses including $k$-demand and SPLC valuations. 2. Ensuring Low Subsidy: We investigate the quantity of subsidy required to guarantee envy-freeness online. Surprisingly, even for additive valuations, the minimum subsidy may be as large as $Ω(mn)$, in contrast to the offline setting, where the bound is $O(n)$. On the positive side, we identify valuation classes where the minimum subsidy is small (i.e., does not depend on $m$), including $k$-valued, rank-one, restricted additive, and identical valuations, and we obtain (mostly) tight subsidy bounds for these classes.

Problem

• Studies fair online allocation of indivisible items among agents
• Investigates maintaining envy-free allocations with minimal subsidy
• Analyzes subsidy bounds across different valuation classes online

Innovation

• Extends envy-free allocation with subsidy to online settings
• Designs online algorithms for additive and related valuations
• Identifies valuation classes requiring small bounded subsidy