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A Complete Landscape for the Price of Envy-Freeness

Zihao Li, Shengxin Liu, Xinhang Lu, Biaoshuai Tao, Yichen Tao

TL;DR

This work delivers a complete landscape of the price of envy-freeness across indivisible and mixed goods, resolving the two-agent EF1 bound at $8/7$ under scaled utilities and providing tight (and asymptotically tight) bounds for EFX, EFM, and EFXM. It extends insights to arbitrary numbers of agents using a discretization-and-limit approach, establishing $Θ(\sqrt{n})$ bounds for scaled utilities and $Θ(n)$ for unscaled utilities. A Cut-and-Choose based method yields EFXM with strong welfare guarantees in the two-agent setting, and the results reveal that EFM and EFXM share the same price as EFX under several scenarios. Overall, the paper clarifies the efficiency-loss trade-offs under envy-freeness constraints and lays groundwork for fair allocation in mixed divisible/indivisible resource environments.

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.

A Complete Landscape for the Price of Envy-Freeness

TL;DR

This work delivers a complete landscape of the price of envy-freeness across indivisible and mixed goods, resolving the two-agent EF1 bound at under scaled utilities and providing tight (and asymptotically tight) bounds for EFX, EFM, and EFXM. It extends insights to arbitrary numbers of agents using a discretization-and-limit approach, establishing bounds for scaled utilities and for unscaled utilities. A Cut-and-Choose based method yields EFXM with strong welfare guarantees in the two-agent setting, and the results reveal that EFM and EFXM share the same price as EFX under several scenarios. Overall, the paper clarifies the efficiency-loss trade-offs under envy-freeness constraints and lays groundwork for fair allocation in mixed divisible/indivisible resource environments.

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 agents, for both scaled and unscaled utilities.
Paper Structure (24 sections, 8 theorems, 26 equations, 3 tables, 2 algorithms)

This paper contains 24 sections, 8 theorems, 26 equations, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Additional Related Work
  3. Preliminaries
  4. Utilities
  5. Allocations
  6. Scaling
  7. Social Welfare
  8. Fairness Notions
  9. Price of Fairness
  10. Partial Allocation and Resource Monotonicity
  11. Two Agents
  12. Price of EF1 (for Indivisible Goods)
  13. Case 1: $y \geq \frac{1}{2}$
  14. Case 2: $y \leq \frac{1}{3}$
  15. Case 3: $\frac{1}{3} \leq y \leq \frac{1}{2}$
  16. ...and 9 more sections

Key Result

Theorem 3.1

For $n = 2$ and scaled utilities (over indivisible goods), the price of EF1 is $\frac{8}{7}$.

Theorems & Definitions (21)

  • Remark 2.1: Piecewise-constant valuation assumption
  • Definition 2.2: EF1 LiptonMaMo04Budish11
  • Definition 2.3: EFX CaragiannisKuMo19PlautRo20
  • Definition 2.4: EFM BeiLiLi21
  • Definition 2.5: EFXM NishimuraSu23
  • Definition 2.6: Price of Fairness $P$
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • proof : Proof of \ref{['thm:indivisible-2-agents']}
  • ...and 11 more