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

## TL;DR

This paper analyzes the balance between envy and efficiency in repeated resource allocations with stochastic replenishments, revealing a phase transition in inefficiency based on storage capacity and envy levels, with implications for supply-driven systems.

## Contribution

It introduces a class of Bang-Bang control policies and characterizes their inefficiency, demonstrating a sharp phase transition and establishing lower bounds for envy-inefficiency trade-offs in replenishment settings.

## Key findings

- Inefficiency drops exponentially with storage capacity when envy is positive.
- Trade-offs are primarily driven by supply variability, not demand.
- Policies exhibit a phase transition in inefficiency based on envy target.

## 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 $\Theta(1/M)$ when $\Delta = 0$ to $e^{-\Omega(\Delta M)}$ when $\Delta > 0$, where $\Delta$ 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.