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Simultaneous Ordinal Maximin Share and Envy-Based Guarantees

Hannaneh Akrami, Timo Reichert

TL;DR

The paper investigates aligning ordinal MMS guarantees with envy-based fairness (EFX and EF1) in indivisible good allocations under additive valuations. It introduces top-$n$ and normalization techniques, along with tailored algorithms, to achieve complete allocations that simultaneously satisfy MMS-based and envy-based criteria in ordered and top-$n$ settings. Specifically, it proves that there exist complete allocations that are 1-out-of-$\lceil 3n/2 \rceil$ MMS with EFX for ordered instances; complete MMS/EF1 allocations for top-$n$ instances; and complete 1-out-of-$4\lceil n/3 \rceil$ MMS with EF1 for ordered instances, using envy-cycle elimination to complete partial solutions. These results advance understanding of how ordinal fairness notions interact with envy-based guarantees and demonstrate that complete allocations combining strong MMS and envy properties are achievable in structured settings, matching or improving state-of-the-art ordinal MMS bounds for ordered cases and extending to broader instance classes.

Abstract

We study the fair allocation of indivisible goods among agents with additive valuations. The fair division literature has traditionally focused on two broad classes of fairness notions: envy-based notions and share-based notions. Within the share-based framework, most attention has been devoted to the maximin share (MMS) guarantee and its relaxations, while envy-based fairness has primarily centered on EFX and its relaxations. Recent work has shown the existence of allocations that simultaneously satisfy multiplicative approximate MMS and envy-based guarantees such as EF1 or EFX. Motivated by this line of research, we study for the first time the compatibility between ordinal approximations of MMS and envy-based fairness notions. In particular, we establish the existence of allocations satisfying the following combined guarantees: (i) simultaneous $1$-out-of-$\lceil 3n/2 \rceil$ MMS and EFX for ordered instances; (ii) simultaneous $1$-out-of-$\lceil 3n/2 \rceil$ MMS and EF1 for top-$n$ instances; and (iii) simultaneous $1$-out-of-$4\lceil n/3 \rceil$ MMS and EF1 for ordered instances.

Simultaneous Ordinal Maximin Share and Envy-Based Guarantees

TL;DR

The paper investigates aligning ordinal MMS guarantees with envy-based fairness (EFX and EF1) in indivisible good allocations under additive valuations. It introduces top- and normalization techniques, along with tailored algorithms, to achieve complete allocations that simultaneously satisfy MMS-based and envy-based criteria in ordered and top- settings. Specifically, it proves that there exist complete allocations that are 1-out-of- MMS with EFX for ordered instances; complete MMS/EF1 allocations for top- instances; and complete 1-out-of- MMS with EF1 for ordered instances, using envy-cycle elimination to complete partial solutions. These results advance understanding of how ordinal fairness notions interact with envy-based guarantees and demonstrate that complete allocations combining strong MMS and envy properties are achievable in structured settings, matching or improving state-of-the-art ordinal MMS bounds for ordered cases and extending to broader instance classes.

Abstract

We study the fair allocation of indivisible goods among agents with additive valuations. The fair division literature has traditionally focused on two broad classes of fairness notions: envy-based notions and share-based notions. Within the share-based framework, most attention has been devoted to the maximin share (MMS) guarantee and its relaxations, while envy-based fairness has primarily centered on EFX and its relaxations. Recent work has shown the existence of allocations that simultaneously satisfy multiplicative approximate MMS and envy-based guarantees such as EF1 or EFX. Motivated by this line of research, we study for the first time the compatibility between ordinal approximations of MMS and envy-based fairness notions. In particular, we establish the existence of allocations satisfying the following combined guarantees: (i) simultaneous -out-of- MMS and EFX for ordered instances; (ii) simultaneous -out-of- MMS and EF1 for top- instances; and (iii) simultaneous -out-of- MMS and EF1 for ordered instances.
Paper Structure (14 sections, 37 theorems, 41 equations, 4 algorithms)

This paper contains 14 sections, 37 theorems, 41 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Technical Contribution
  4. Further Related Work
  5. Definitions and Notation
  6. $1$-out-of-$\lceil 3n/2 \rceil$ MMS Together with EFX for Ordered Instances
  7. $1$-out-of-$\lceil 3n/2 \rceil$ MMS Together with EF1 for Top-$n$ Instances
  8. $1$-out-of-$4\lceil n/3 \rceil$ MMS Together with EF1 for Ordered Instances
  9. Working with Normalization
  10. The Algorithm
  11. Conclusion
  12. Missing Proofs of Section \ref{['sec2']}
  13. Missing Proofs of Section \ref{['sec4']}
  14. Missing Proofs of Section \ref{['sec5']}

Key Result

Theorem 1

For every ordered instance $(M,N,(v_i)_{i\in N})$ of fair division, there exists a complete allocation that is 1-out-of-$\lceil 3n/2 \rceil$ MMS and EFX at the same time.

Theorems & Definitions (80)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • ...and 70 more