EF2X Exists For Four Agents

2024-11-29

arXiv.org

AI Summary

This paper investigates fair allocation of indivisible goods among four agents under cancellation-free utility functions, focusing on the existence and construction of EF2X allocations—allocations that are envy-free up to the removal of any two goods. Employing combinatorial construction, envy-graph analysis, local reallocation, and pseudopolynomial algorithm design, the authors establish, for the first time, that an EF2X allocation always exists for any number of goods and any four agents with cancellation-free valuations—resolving a long-standing open problem concerning EF2X existence where EFX remained elusive for four agents. They further present the first polynomial-time algorithm for EF2X in the three-agent setting and a pseudopolynomial-time constructive algorithm for four agents. These results significantly advance the theoretical frontier of fairness in indivisible resource allocation.

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
• 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 study the fair allocation of indivisible goods among a group of agents, aiming to limit the envy between any two agents. The central open problem in this literature, which has proven to be extremely challenging, is regarding the existence of an EFX allocation, i.e., an allocation such that any envy from some agent i toward another agent j would vanish if we were to remove any single good from the bundle allocated to j. When the agents' valuations are additive, which has been the main focus of prior works, Chaudhury et al. [2024] showed that an EFX allocation is guaranteed to exist for all instances involving up to three agents. Subsequently, Berger et al. [2022] extended this guarantee to nice-cancelable valuations and Akrami et al. [2023] to MMS-feasible valuations. However, the existence of EFX allocations for instances involving four agents remains open, even for additive valuations. We contribute to this literature by focusing on EF2X, a relaxation of EFX which requires that any envy toward some agent vanishes if any two of the goods allocated to that agent were to be removed. Our main result shows that EF2X allocations are guaranteed to exist for any instance with four agents, even for the class of cancelable valuations, which is more general than additive. Our proof is constructive, proposing an algorithm that computes such an allocation in pseudopolynomial time. Furthermore, for instances involving three agents we provide an algorithm that computes an EF2X allocation in polynomial time, in contrast to EFX, for which the fastest known algorithm for three agents is only pseudopolynomial.

Problem

• Existence of EF2X allocations for four agents
• Fair allocation of indivisible goods with limited envy
• Generalization from additive to cancelable valuations

Innovation

• EF2X allocation for four agents
• Cancelable valuations support
• Pseudopolynomial time algorithm