EFX Exists for Three Types of Agents

2024-10-17

arXiv.org

AI Summary

This paper addresses the long-standing open problem of existence of envy-free up to any good (EFX) allocations for indivisible goods. While prior work established EFX existence only for at most three agents or for settings with at most two distinct additive valuation types, the general case remained unresolved. We prove, for the first time, that an EFX allocation always exists when agents’ additive valuations belong to at most three distinct types—regardless of the number of agents. This result unifies and strictly generalizes previous results for three agents and for two valuation types, thereby resolving the existence question in this broad setting. Technically, our proof integrates tools from combinatorial game theory, inductive construction, valuation-type classification, and pairwise reallocation strategies, yielding the first existence framework for EFX that accommodates arbitrarily many agents under a bounded number (≤3) of valuation types—filling a fundamental gap in fair division theory.

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: Large-scale security measurements

Abstract

We study the problem of finding an envy-free allocation of indivisible goods among agents with additive valuations. We focus on the fairness notion of envy-freeness up to any good (EFX). A central open question in fair division is whether EFX allocations always exist for any number of agents. While EFX has been established for three agents [CGM24] and for any number of agents with at most two distinct valuations [Mah23], its existence in more general settings remains open. In this paper, we make significant progress by proving that EFX allocations exist for any number of agents when there are at most three distinct additive valuations. This result simultaneously generalizes both the three-agent case and the two-type case, settling an open question in the field (see [Mah23]).

Problem

• Existence of EFX allocations for multiple agents
• Generalizing EFX to three distinct additive valuations
• Settling open questions in fair division

Innovation

• EFX allocations for three distinct valuations
• Generalizes three-agent and two-type cases
• Settles open question in fair division