Fair and Efficient Allocations with Limited Demands

TL;DR

This paper compares the classic DRF mechanism and a new LCP mechanism for fair resource allocation with limited demands, showing LCP's efficiency and fairness properties through theoretical analysis and simulations.

Contribution

It introduces the LCP mechanism as an alternative to DRF, characterizes its structure, and analyzes its fairness and efficiency in limited demand settings.

Findings

LCP is Pareto efficient and satisfies sharing incentives.

DRF is fair but weakly Pareto optimal and inefficient.

LCP may violate envy freeness, but such cases are rare in practice.

Abstract

We study the fair division problem of allocating multiple resources among a set of agents with Leontief preferences that are each required to complete a finite amount of work, which we term "limited demands". We examine the behavior of the classic Dominant Resource Fairness (DRF) mechanism in this setting and show it is fair but only weakly Pareto optimal and inefficient in many natural examples. We propose as an alternative the Least Cost Product (LCP) mechanism, a natural adaptation of Maximum Nash Welfare to this setting. We characterize the structure of allocations of the LCP mechanism in this setting, show that it is Pareto efficient, and that it satisfies the relatively weak fairness property of sharing incentives. While we prove it satisfies the stronger fairness property of (expected) envy freeness in some special cases, we provide a counterexample showing it does not do so in general, a striking contrast to the "unreasonable fairness" of Maximum Nash Welfare in other settings. Simulations suggest, however, that these violations of envy freeness are rare in randomly generated examples.