# Fairness and Utilization in Allocating Resources with Uncertain Demand

## TL;DR

This paper investigates the trade-offs between fairness and utilization in resource allocation under uncertain demand, providing bounds on the gap and identifying distribution families where fairness and maximum utilization can coincide.

## Contribution

It introduces the first provable bounds on the fairness-utilization gap for various demand distributions, including cases with no gap and bounded gaps for power-law distributions.

## Key findings

- For arbitrary distributions, the gap can be bounded.
- Exponential and Weibull distributions can achieve maximum utilization with fairness.
- Power-law distributions have a bounded, non-trivial gap.

## Abstract

Resource allocation problems are a fundamental domain in which to evaluate the fairness properties of algorithms. The trade-offs between fairness and utilization have a long history in this domain. A recent line of work has considered fairness questions for resource allocation when the demands for the resource are distributed across multiple groups and drawn from probability distributions. In such cases, a natural fairness requirement is that individuals from different groups should have (approximately) equal probabilities of receiving the resource. A largely open question in this area has been to bound the gap between the maximum possible utilization of the resource and the maximum possible utilization subject to this fairness condition. Here, we obtain some of the first provable upper bounds on this gap. We obtain an upper bound for arbitrary distributions, as well as much stronger upper bounds for specific families of distributions that are typically used to model levels of demand. In particular, we find - somewhat surprisingly - that there are natural families of distributions (including Exponential and Weibull) for which the gap is non-existent: it is possible to simultaneously achieve maximum utilization and the given notion of fairness. Finally, we show that for power-law distributions, there is a non-trivial gap between the solutions, but this gap can be bounded by a constant factor independent of the parameters of the distribution.