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New Fairness Concepts for Allocating Indivisible Items

Ioannis Caragiannis, Jugal Garg, Nidhi Rathi, Eklavya Sharma, Giovanna Varricchio

TL;DR

This paper introduces two robust fairness notions, Epistemic EFX (EEFX) and Minimum EFX Share (MXS), to address the limitations of envy-freeness up to any item (EFX) and maximin fairness (MMS) for allocating indivisible items. It proves that EEFX allocations always exist and can be computed in polynomial time for additive valuations, while establishing a chain of implications MMS -> EEFX -> MXS -> PROP1 that places these concepts within the familiar fairness landscape. The work also shows that computing the minimum EFX share is NP-hard, yet the proposed algorithm yields EEFX (hence MXS) allocations and even 2/3-MMS as a bonus, extending to chores and to cancelable valuations with certificates. The results provide a practical and theoretically grounded alternative to EFX and MMS, with extensions to broader valuation classes and notable implications for privacy-aware or large-scale fair division scenarios.

Abstract

For the fundamental problem of fairly dividing a set of indivisible items among agents, envy-freeness up to any item (EFX) and maximin fairness (MMS) are arguably the most compelling fairness concepts proposed until now. Unfortunately, despite significant efforts over the past few years, whether EFX allocations always exist is still an enigmatic open problem, let alone their efficient computation. Furthermore, today we know that MMS allocations are not always guaranteed to exist. These facts weaken the usefulness of both EFX and MMS, albeit their appealing conceptual characteristics. We propose two alternative fairness concepts, called epistemic EFX (EEFX) and minimum EFX share fairness (MXS), inspired by EFX and MMS. For both, we explore their relationships to well-studied fairness notions and, more importantly, prove that EEFX and MXS allocations always exist and can be computed efficiently for additive valuations. Our results justify that the new fairness concepts can be excellent alternatives to EFX and MMS.

New Fairness Concepts for Allocating Indivisible Items

TL;DR

This paper introduces two robust fairness notions, Epistemic EFX (EEFX) and Minimum EFX Share (MXS), to address the limitations of envy-freeness up to any item (EFX) and maximin fairness (MMS) for allocating indivisible items. It proves that EEFX allocations always exist and can be computed in polynomial time for additive valuations, while establishing a chain of implications MMS -> EEFX -> MXS -> PROP1 that places these concepts within the familiar fairness landscape. The work also shows that computing the minimum EFX share is NP-hard, yet the proposed algorithm yields EEFX (hence MXS) allocations and even 2/3-MMS as a bonus, extending to chores and to cancelable valuations with certificates. The results provide a practical and theoretically grounded alternative to EFX and MMS, with extensions to broader valuation classes and notable implications for privacy-aware or large-scale fair division scenarios.

Abstract

For the fundamental problem of fairly dividing a set of indivisible items among agents, envy-freeness up to any item (EFX) and maximin fairness (MMS) are arguably the most compelling fairness concepts proposed until now. Unfortunately, despite significant efforts over the past few years, whether EFX allocations always exist is still an enigmatic open problem, let alone their efficient computation. Furthermore, today we know that MMS allocations are not always guaranteed to exist. These facts weaken the usefulness of both EFX and MMS, albeit their appealing conceptual characteristics. We propose two alternative fairness concepts, called epistemic EFX (EEFX) and minimum EFX share fairness (MXS), inspired by EFX and MMS. For both, we explore their relationships to well-studied fairness notions and, more importantly, prove that EEFX and MXS allocations always exist and can be computed efficiently for additive valuations. Our results justify that the new fairness concepts can be excellent alternatives to EFX and MMS.
Paper Structure (20 sections, 31 theorems, 32 equations, 3 tables, 3 algorithms)

This paper contains 20 sections, 31 theorems, 32 equations, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our Conceptual and Technical Contribution
  3. Definitions and Notation
  4. Notions of Fairness
  5. Ordering
  6. Relations to Other Fairness Concepts
  7. Existence and Efficient Computation of EEFX Allocations
  8. Computing the Minimum EFX Share
  9. Chores
  10. Relations to Other Fairness Concepts
  11. Cancelable Valuations
  12. Comparing Bundles using Ordered Cancelable Valuations
  13. EEFX for Cancelable Valuations
  14. Discussion
  15. Defining EFX for Non-Additive Valuations
  16. ...and 5 more sections

Key Result

Theorem 1

An EEFX allocation in a fair division instance is also MXS.

Theorems & Definitions (78)

  • Example 1
  • Example 2
  • Definition 1: Envy-freeness up to any item (EFX)
  • Definition 2: Epistemic EFX (EEFX) and EEFX certificates
  • Definition 3: Maximin share (MMS)
  • Definition 4: Minimum EFX share, MXS allocations
  • Definition 5: Proportionality up to one item (PROP1)
  • Claim 1
  • Theorem 1: ${\normalfont\textsc{EEFX}} \Rightarrow {\normalfont\textsc{MXS}}$
  • proof
  • ...and 68 more