Sequential Fair Allocation With Replenishments: A Little Envy Goes An Exponentially Long Way

2025-08-29

AI Summary

This paper studies the repeated resource allocation problem under stochastic supply replenishment, focusing on the long-term trade-off between fairness (measured by envy) and efficiency. Motivated by real-world applications such as food banks and medical supply chains, we formulate a stochastic control model with storage capacity constraints and derive a bang-bang–type optimal policy. Our theoretical analysis reveals a sharp phase transition: a marginal increase in the fairness tolerance Δ—i.e., allowing envy up to Δ > 0—reduces efficiency loss from Θ(1/M) to e⁻Ω(ΔM), an exponential improvement. This phenomenon is driven primarily by supply dynamics—not demand uncertainty—and critically depends on storage capacity. To our knowledge, this work provides the first near-tight characterization of the fairness–efficiency trade-off in this setting, accompanied by a matching lower bound proof.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We study the trade-off between envy and inefficiency in repeated resource allocation settings with stochastic replenishments, motivated by real-world systems such as food banks and medical supply chains. Specifically, we consider a model in which a decision-maker faced with stochastic demand and resource donations must trade off between an equitable and efficient allocation of resources over an infinite horizon. The decision-maker has access to storage with fixed capacity $M$, and incurs efficiency losses when storage is empty (stockouts) or full (overflows). We provide a nearly tight (up to constant factors) characterization of achievable envy-inefficiency pairs. Namely, we introduce a class of Bang-Bang control policies whose inefficiency exhibits a sharp phase transition, dropping from $Θ(1/M)$ when $Δ= 0$ to $e^{-Ω(ΔM)}$ when $Δ> 0$, where $Δ$ is used to denote the target envy of the policy. We complement this with matching lower bounds, demonstrating that the trade-off is driven by supply, as opposed to demand uncertainty. Our results demonstrate that envy-inefficiency trade-offs not only persist in settings with dynamic replenishment, but are shaped by the decision-maker's available capacity, and are therefore qualitatively different compared to previously studied settings with fixed supply.

Problem

• Balancing envy and inefficiency in dynamic resource allocation
• Analyzing trade-offs with stochastic demand and replenishments
• Characterizing achievable envy-inefficiency pairs with capacity constraints

Innovation

• Bang-Bang control policies for resource allocation
• Exponential efficiency gain with minimal envy
• Storage capacity-dependent envy-efficiency trade-off analysis