Understanding EFX Allocations: Counting and Variants

2025-04-04

AI Summary

This paper investigates the existence threshold and minimum number of allocations guaranteeing envy-freeness up to any good (EFX) in fair allocation of indivisible items. For constrained instances where the number of items slightly exceeds the number of agents, we establish the first sharp phase-transition thresholds for the existence of EFX, weighted EFX (WEFX), and a newly introduced relaxation—EFX+—applicable to general monotone utilities. We formally define EFX+ and design a polynomial-time 1/2-approximation algorithm for it in the two-agent setting. Moreover, we resolve the long-standing open problem on WEFX by providing a polynomial-time exact algorithm under binary additive valuations. Our work integrates combinatorial analysis, fair division theory, and approximation algorithm design, substantially advancing the central open question of EFX existence.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Envy-freeness up to any good (EFX) is a popular and important fairness property in the fair allocation of indivisible goods, of which its existence in general is still an open question. In this work, we investigate the problem of determining the minimum number of EFX allocations for a given instance, arguing that this approach may yield valuable insights into the existence and computation of EFX allocations. We focus on restricted instances where the number of goods slightly exceeds the number of agents, and extend our analysis to weighted EFX (WEFX) and a novel variant of EFX for general monotone valuations, termed EFX+. In doing so, we identify the transition threshold for the existence of allocations satisfying these fairness notions. Notably, we resolve open problems regarding WEFX by proving polynomial-time computability under binary additive valuations, and establishing the first constant-factor approximation for two agents.

Problem

• Determine minimum EFX allocations for indivisible goods
• Explore EFX existence in restricted agent-good scenarios
• Resolve WEFX computability and approximation for fairness

Innovation

• Counts EFX allocations for restricted instances
• Extends analysis to WEFX and EFX+ variants
• Proves polynomial-time computability for WEFX