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Simultaneously Satisfying MXS and EFL

Arash Ashuri, Vasilis Gkatzelis

TL;DR

This work addresses the incompatibility of two core fairness concepts for indivisible goods by proving the existence of allocations that simultaneously satisfy MXS (a relaxation of proportionality) and EFL (a relaxation of envy-freeness) for the class of monotone restricted-MMS-feasible valuations, which extends additive settings. It introduces a constructive two-phase ReBalance algorithm that iteratively builds partitions and full MXS+EFL associations, leveraging Generalized Envy Graphs, Envy Cycle Elimination, and invariants to guarantee termination. In the additive case, these allocations imply strong guarantees such as $4/7$-MMS for MXS and EF1, $1/2$-GMMS, $1/2$-EFX, and $2/3$-PMMS for EFL, yielding a spectrum of fairness properties that hold simultaneously. The results advance the understanding of universal fairness in discrete resource allocation and provide a broadly applicable algorithm for achieving robust combinations of fairness notions with practical implications for applications requiring fair division under indivisibility constraints.

Abstract

The two standard fairness notions in the resource allocation literature are proportionality and envy-freeness. If there are n agents competing for the available resources, then proportionality requires that each agent receives at least a 1/n fraction of their total value for the set of resources. On the other hand, envy-freeness requires that each agent weakly prefers the resources allocated to them over those allocated to any other agent. Each of these notions has its own benefits, but it is well known that neither one of the two is always achievable when the resources being allocated are indivisible. As a result, a lot of work has focused on satisfying fairness notions that relax either proportionality or envy-freeness. In this paper, we focus on MXS (a relaxation of proportionality) and EFL (a relaxation of envy-freeness). Each of these notions was previously shown to be achievable on its own [Barman et al.,2018, Caragiannis et al., 2023], and our main result is an algorithm that computes allocations that simultaneously satisfy both, combining the benefits of approximate proportionality and approximate envy-freeness. In fact, we prove this for any instance involving agents with valuation functions that are restricted MMS-feasible, which are more general than additive valuations. Also, since every EFL allocation directly satisfies other well-studied fairness notions like EF1, 1/2-EFX, 1/2-GMMS, and 2/3-PMMS, and every MXS allocation satisfies 4/7-MMS, the allocations returned by our algorithm simultaneously satisfy a wide variety of fairness notions and are, therefore, universally fair [Amanatidis et al., 2020].

Simultaneously Satisfying MXS and EFL

TL;DR

This work addresses the incompatibility of two core fairness concepts for indivisible goods by proving the existence of allocations that simultaneously satisfy MXS (a relaxation of proportionality) and EFL (a relaxation of envy-freeness) for the class of monotone restricted-MMS-feasible valuations, which extends additive settings. It introduces a constructive two-phase ReBalance algorithm that iteratively builds partitions and full MXS+EFL associations, leveraging Generalized Envy Graphs, Envy Cycle Elimination, and invariants to guarantee termination. In the additive case, these allocations imply strong guarantees such as -MMS for MXS and EF1, -GMMS, -EFX, and -PMMS for EFL, yielding a spectrum of fairness properties that hold simultaneously. The results advance the understanding of universal fairness in discrete resource allocation and provide a broadly applicable algorithm for achieving robust combinations of fairness notions with practical implications for applications requiring fair division under indivisibility constraints.

Abstract

The two standard fairness notions in the resource allocation literature are proportionality and envy-freeness. If there are n agents competing for the available resources, then proportionality requires that each agent receives at least a 1/n fraction of their total value for the set of resources. On the other hand, envy-freeness requires that each agent weakly prefers the resources allocated to them over those allocated to any other agent. Each of these notions has its own benefits, but it is well known that neither one of the two is always achievable when the resources being allocated are indivisible. As a result, a lot of work has focused on satisfying fairness notions that relax either proportionality or envy-freeness. In this paper, we focus on MXS (a relaxation of proportionality) and EFL (a relaxation of envy-freeness). Each of these notions was previously shown to be achievable on its own [Barman et al.,2018, Caragiannis et al., 2023], and our main result is an algorithm that computes allocations that simultaneously satisfy both, combining the benefits of approximate proportionality and approximate envy-freeness. In fact, we prove this for any instance involving agents with valuation functions that are restricted MMS-feasible, which are more general than additive valuations. Also, since every EFL allocation directly satisfies other well-studied fairness notions like EF1, 1/2-EFX, 1/2-GMMS, and 2/3-PMMS, and every MXS allocation satisfies 4/7-MMS, the allocations returned by our algorithm simultaneously satisfy a wide variety of fairness notions and are, therefore, universally fair [Amanatidis et al., 2020].

Paper Structure

This paper contains 26 sections, 44 theorems, 7 equations, 1 figure, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Preliminaries
  4. Types of Valuations.
  5. Envy, EFX-envy, and EFL-envy.
  6. EFX, EFL, EEFX, and MXS
  7. Association Function s and Allocations.
  8. Generalized Envy Graph.
  9. Envy Cycle Elimination.
  10. Chains and subchains.
  11. Shifting a subchain.
  12. Algorithm Constructing MXS+EFL Allocations
  13. Subroutines Used in ReBalance
  14. Some Additional Notation.
  15. Some Useful Lemmas and Observations
  16. ...and 11 more sections

Key Result

Theorem 3.1

If all the valuations are monotone and restricted-MMS-feasible, MXS+EFL always terminates and returns an allocation that is simultaneously MXS and EFL for every agent.

Figures (1)

  • Figure 1: This figure demonstrates how association function changes after ShiftSubChain. Dashed arrows show agents associated with bundles.

Theorems & Definitions (100)

  • Definition 2.1: restricted MMS feasible
  • Definition 2.2: Set of chains
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 5.1
  • ...and 90 more