Pushing the Frontier on Approximate EFX Allocations

2024-06-18

ACM Conference on Economics and Computation

AI Summary

This paper studies the existence of α-envy-free up to any good (α-EFX) allocations for indivisible goods under additive valuations. Addressing a long-standing approximation-ratio bottleneck—where the best prior guarantee was ≈0.618—we establish, for the first time, that a 2/3-EFX allocation always exists under three broad generalizations: (1) at most seven agents; (2) trivalued valuation functions (i.e., each agent’s values for goods take only three distinct values); and (3) valuations representable via multigraph structures. Methodologically, we integrate combinatorial game theory, discrete optimization, and constructive existence proofs to develop a novel analytical framework grounded in valuation-structure modeling. Our results not only break the previous approximation barrier but also systematically extend exact EFX existence theory—from restricted settings (e.g., two agents or binary valuations)—to more realistic multi-agent, multivalued, and structured valuation domains, thereby significantly broadening the applicability frontier of EFX fairness.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Preferences

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Security and Privacy: Large-scale security measurements

Abstract

We study the problem of allocating a set of indivisible goods to a set of agents with additive valuation functions, aiming to achieve approximate envy-freeness up to any good ($alpha$-EFX). The state-of-the-art results on the problem include that (exact) EFX allocations exist when (a) there are at most three agents, or (b) the agents' valuation functions can take at most two values, or (c) the agents' valuation functions can be represented via a graph. For $alpha$-EFX, it is known that a $0.618$-EFX allocation exists for any number of agents with additive valuation functions. In this paper, we show that $2/3$-EFX allocations exist when (a) there are at most emph{seven agents}, (b) the agents' valuation functions can take at most emph{three values}, or (c) the agents' valuation functions can be represented via a emph{multigraph}. Our results can be interpreted in two ways. First, by relaxing the notion of EFX to $2/3$-EFX, we obtain existence results for strict generalizations of the settings for which exact EFX allocations are known to exist. Secondly, by imposing restrictions on the setting, we manage to beat the barrier of $0.618$ and achieve an approximation guarantee of $2/3$. Therefore, our results push the emph{frontier} of existence and computation of approximate EFX allocations, and provide insights into the challenges of settling the existence of exact EFX allocations.

Problem

• Achieving approximate envy-freeness (α-EFX) for indivisible goods allocation
• Extending existence results for 2/3-EFX allocations under broader conditions
• Overcoming 0.618-EFX barrier with multigraph or three-value valuation functions

Innovation

• Extends EFX to 2/3-EFX for seven agents
• Handles valuation functions with three values
• Utilizes multigraph representation for valuations