Achieving EF1 and Epistemic EFX Guarantees Simultaneously

2026-02-12

AI Summary

This work addresses the open question posed by Akrami and Rathi (2025) regarding the coexistence of EF1 and epistemically envy-free up to any good (EEFX) in the allocation of indivisible goods under additive valuations. We introduce the novel notion of a “strong EEFX share” and, through a combination of combinatorial arguments and mechanism design techniques, provide the first constructive proof that there always exists an allocation satisfying both EF1—and even the stronger EFL—and EEFX under additive valuations. Our result not only establishes the compatibility of EF1 and EEFX but also represents significant progress toward resolving the long-standing open problem of the existence of EFX allocations.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Preferences

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Privacy-enhancing technologies

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.

Problem

• EFX
• EF1
• fair division
• indivisible goods
• epistemic EFX

Innovation

• EFX
• EF1
• epistemic EFX
• fair division
• additive valuations