A Complete Landscape for the Price of Envy-Freeness

TL;DR

This paper provides a comprehensive analysis of the price of envy-freeness in fair division, establishing tight bounds for various fairness notions and settings, including two-agent and mixed goods scenarios.

Contribution

It offers the first complete characterization of the price of envy-freeness for multiple fairness notions across different settings, resolving open questions in the literature.

Findings

Tight bounds for the price of EF1 and EFX in two-agent indivisible goods setting.

Complete characterization of the price of EFM and EFXM in mixed goods scenarios.

Asymptotically tight bounds for multiple agents and utility types.

Abstract

We study the efficiency of fair allocations using the well-studied price of fairness concept, which quantitatively measures the worst-case efficiency loss when imposing fairness constraints. Previous works provided partial results on the price of fairness with well-known fairness notions such as envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX). In this paper, we give a complete characterization for the price of envy-freeness in various settings. In particular, we first consider the two-agent case under the indivisible-goods setting and present tight ratios for the price of EF1 (for scaled utility) and EFX (for unscaled utility), which resolve questions left open in the literature. Next, we consider the mixed goods setting which concerns a mixture of both divisible and indivisible goods. We focus on envy-freeness for mixed goods (EFM), which generalizes both envy-freeness and EF1, as well as its strengthening called envy-freeness up to any good for mixed goods (EFXM), which generalizes envy-freeness and EFX. To this end, we settle the price of EFM and EFXM by providing a complete picture of tight bounds for two agents and asymptotically tight bounds for $n$ agents, for both scaled and unscaled utilities.