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Achieving EF1 and Epistemic EFX Guarantees Simultaneously

Hannaneh Akrami, Ryoga Mahara, Kurt Mehlhorn, Nidhi Rathi

TL;DR

The paper resolves a major open problem by proving that, for additive valuations, there exists an allocation that is simultaneously $EF1$ (indeed $EFL$) and $EEFX$, thus bridging envy-based and epistemic fairness notions. The authors introduce the strong $EEFX$ share $\theta_i$, show $\theta_i \le RMMS(M,v_i,n)$, and use this to construct a complete allocation that is $EFL$ and $EEFX$, hence $EF1$ as well. This result relies on the lone-divider framework and a careful analysis of $EEFX$ feasibility through share-based thresholds, with $\theta_i$ strictly larger than the prior $MXS_i$ in some cases. The work situates itself among MMS and RMMS literature, showing that aligning share-based guarantees with envy-based relaxations can yield robust simultaneous guarantees, while leaving computational and broader-valuation extensions as important future directions.

Abstract

We study the fundamental problem of fairly dividing a set of indivisible goods among agents with additive valuations. Here, envy-freeness up to any good (EFX) is a central fairness notion and resolving its existence is regarded as one of the most important open problems in this area of research. Two prominent relaxations of EFX are envy-freeness up to one good (EF1) and epistemic EFX (EEFX). While allocations satisfying each of these notions individually are known to exist even for general monotone valuations, whether both can be satisfied simultaneously remains open for all instances in which the EFX problem is itself unresolved. In this work, we show that there always exists an allocation that is both EF1 (in fact, the stronger notion EFL) and EEFX for additive valuations, thereby resolving the primary open question raised by Akrami and Rathi (2025) and bringing us one step closer to resolving the elusive EFX problem. We introduce a new share-based fairness notion, termed strong EEFX share, which may be of independent interest and which implies EEFX feasibility of bundles. We show that this notion is compatible with EF1, leading to the desired existence result.

Achieving EF1 and Epistemic EFX Guarantees Simultaneously

TL;DR

The paper resolves a major open problem by proving that, for additive valuations, there exists an allocation that is simultaneously (indeed ) and , thus bridging envy-based and epistemic fairness notions. The authors introduce the strong share , show , and use this to construct a complete allocation that is and , hence as well. This result relies on the lone-divider framework and a careful analysis of feasibility through share-based thresholds, with strictly larger than the prior in some cases. The work situates itself among MMS and RMMS literature, showing that aligning share-based guarantees with envy-based relaxations can yield robust simultaneous guarantees, while leaving computational and broader-valuation extensions as important future directions.

Abstract

We study the fundamental problem of fairly dividing a set of indivisible goods among agents with additive valuations. Here, envy-freeness up to any good (EFX) is a central fairness notion and resolving its existence is regarded as one of the most important open problems in this area of research. Two prominent relaxations of EFX are envy-freeness up to one good (EF1) and epistemic EFX (EEFX). While allocations satisfying each of these notions individually are known to exist even for general monotone valuations, whether both can be satisfied simultaneously remains open for all instances in which the EFX problem is itself unresolved. In this work, we show that there always exists an allocation that is both EF1 (in fact, the stronger notion EFL) and EEFX for additive valuations, thereby resolving the primary open question raised by Akrami and Rathi (2025) and bringing us one step closer to resolving the elusive EFX problem. We introduce a new share-based fairness notion, termed strong EEFX share, which may be of independent interest and which implies EEFX feasibility of bundles. We show that this notion is compatible with EF1, leading to the desired existence result.
Paper Structure (15 sections, 11 theorems, 18 equations, 6 figures, 2 algorithms)

This paper contains 15 sections, 11 theorems, 18 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Envy-Based Notions
  3. Share-Based Notions
  4. Lone Divider Approach
  5. Technical Contribution
  6. Further Related Work
  7. Preliminaries
  8. Non-degeneracy
  9. Existence of an Allocation Satisfying EFL and EEFX for Additive Valuations
  10. Future Directions
  11. Monotone valuations.
  12. Chores.
  13. Complexity of finding EF1 + EEFX allocations.
  14. Strong EEFX share vs. MMS
  15. Pseudocodes

Key Result

Theorem 1.1

Given a fair division instance $\mathcal{I} = (N,M,\{v_i\}_{i \in N})$ with additive valuation functions, there exists an allocation $X$ that is simultaneously EFL and EEFX.

Figures (6)

  • Figure 1: Relationships among fairness notions under additive valuations.
  • Figure 2: Visualization of the goods in \ref{['example']} from the point of view of agent $1$, using the same color coding. This partition shows that $X_1=\{g_1,g_2\}$ is EEFX feasible.
  • Figure 3: Visualization of the bundles in \ref{['lem:lone-divider']}.
  • Figure 4: $R = T \cap S$ is the union of the red parts and $U = T \setminus S$ is the blue part.
  • Figure 5: Depiction of $S = \bigcup_{i \in [k]} S_i = \bigcup_{i \in [k]} Y_i$, and $R = S \cap T$ in Lemma \ref{['lem1']}.
  • ...and 1 more figures

Theorems & Definitions (29)

  • Theorem 1.1
  • Definition 2.1: EF1 (Envy-freeness up to one good) budish2011combinatorial
  • Definition 2.2: EFL (Envy-freeness up to one less-preferred good) barman2018groupwise
  • Definition 2.3: EFX (Envy-freeness up to any good) caragiannis2019unreasonable
  • Definition 2.4: EFX Partition
  • Theorem 2.5: plaut2020almost
  • Definition 2.6: EEFX feasibility
  • Definition 2.7: EEFX (Epistemic envy-freeness up to any good) Caragiannis2023
  • Definition 2.8: MXS (Minimum EFX share) Caragiannis2023
  • Definition 2.9: Strong EEFX Share
  • ...and 19 more