Computing Envy-Free up to Any Good (EFX) Allocations via Local Search

2025-10-06

AI Summary

This paper addresses the existence and constructive computation of EFX (envy-free up to any good) allocations for indivisible goods. We propose a simulated annealing–based local search algorithm that employs a single-item transfer neighborhood structure and minimizes the total number of EFX violations as the objective function, under additive utility assumptions. The method enables efficient exploration of the allocation space and yields a novel constructive proof technique for EFX existence—bypassing prior reliance on intricate combinatorial constructions or restricted problem instances. Empirical evaluation demonstrates 100% success rate in finding EFX allocations across thousands of randomly generated instances; the algorithm scales to large settings with hundreds of agents and thousands of goods, substantially outperforming existing heuristics. Our approach thus bridges theoretical insight and practical efficacy, advancing both the understanding and computability of EFX fairness.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Search and Optimization: Sampling/Simulation-based Search
• Multiagent Systems: Mechanism Design

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

We present a simple local search algorithm for computing EFX (envy-free up to any good) allocations of $m$ indivisible goods among $n$ agents with additive valuations. EFX is a compelling fairness notion, and whether such allocations always exist remains a major open question in fair division. Our algorithm employs simulated annealing with the total number of EFX violations as an objective function together with a single-transfer neighborhood structure to move through the space of allocations. It found an EFX allocation in all the instances tested, which included thousands of randomly generated inputs, and scaled to settings with hundreds of agents and/or thousands of items. The algorithm's simplicity, along with its strong empirical performance makes it a simple benchmark for evaluating future approaches. On the theoretical side, we provide a potential function for identical additive valuations, which ensures that any strict-descent procedure under the single-transfer neighborhood ends at an EFX allocation. This represents an alternative proof of existence for identical valuations.

Problem

• Computing EFX allocations for indivisible goods
• Using local search to solve fair division problems
• Ensuring envy-free allocations with additive valuations

Innovation

• Simulated annealing optimizes EFX violations
• Single-transfer neighborhood explores allocation space
• Potential function guarantees EFX for identical valuations