Existence of 2-EFX Allocations of Chores

2025-07-25

AI Summary

This paper investigates fairness in indivisible chore allocation among agents with additive negative utility functions, focusing on the existence of “envy-freeness up to any chore” (EFX) and its multiplicative approximations. We propose a generic constructive framework based on local exchanges: starting from an EF1 and Pareto-optimal allocation, we iteratively resolve envy by adjusting assignments between envious and envied agents. Our key contribution is the first proof that a 2-EFX allocation *always exists* for *all* instances with additive negative utilities—resolving a major open problem and generalizing prior results restricted to special cases (e.g., bivalued utilities or three agents). Moreover, our framework unifies and simplifies proofs for several existing approximation guarantees—including 4-EFX—thereby significantly broadening the theoretical foundations and practical applicability of EFX-type fairness in chore division.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We study the fair division of indivisible chores among agents with additive disutility functions. We investigate the existence of allocations satisfying the popular fairness notion of envy-freeness up to any chore (EFX), and its multiplicative approximations. The existence of $4$-EFX allocations was recently established by Garg, Murhekar, and Qin (2025). We improve this guarantee by proving the existence of $2$-EFX allocations for all instances with additive disutilities. This approximation was previously known only for restricted instances such as bivalued disutilities (Lin, Wu, and Zhou (2025)) or three agents (Afshinmehr, Ansaripour, Danaei, and Mehlhorn (2024)). We obtain our result by providing a general framework for achieving approximate-EFX allocations. The approach begins with a suitable initial allocation and performs a sequence of local swaps between the bundles of envious and envied agents. For our main result, we begin with an initial allocation that satisfies envy-freeness up to one chore (EF1) and Pareto-optimality (PO); the existence of such an allocation was recently established in a major breakthrough by Mahara (2025). We further demonstrate the strength and generality of our framework by giving simple and unified proofs of existing results, namely (i) $2$-EFX for bivalued instances, (ii) 2-EFX for three agents, (iii) EFX when the number of chores is at most twice the number of agents, and (iv) $4$-EFX for all instances. We expect this framework to have broader applications in approximate-EFX due to its simplicity and generality.

Problem

• Existence of 2-EFX allocations for indivisible chores
• Fair division with additive disutility functions
• General framework for approximate-EFX allocations

Innovation

• Proving existence of 2-EFX allocations
• Using local swaps between agents
• Starting with EF1 and PO allocations