Almost and Approximate EFX for Few Types of Agents

2025-08-21

AI Summary

This paper studies approximate envy-freeness up to any good (α-EFX) and charitable EFX allocations for indivisible goods when agents have at most $k$ distinct additive valuations. Using combinatorial construction and fair division theory—leveraging structural properties of additive utilities—the authors establish two key results: First, a $2/3$-EFX allocation exists for any number of agents whenever $k leq 4$. Second, under the charitable relaxation—where some goods may remain unallocated—they improve the upper bound on the number of unassigned goods to $widetilde{O}(sqrt{k/varepsilon})$, eliminating dependence on the total number of agents and significantly improving prior bounds. These results represent the first progress on both approximation guarantees and charitable cost in the small-type setting, offering a new paradigm for fair allocation under structured preferences.

Technology Category

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

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 problem of fair allocation of a set of indivisible goods among $n$ agents with $k$ distinct additive valuations, with the goal of achieving approximate envy-freeness up to any good ($α-mathrm{EFX}$). It is known that EFX allocations exist for $n$ agents when there are at most three distinct valuations due to HV et al. Furthermore, Amanatidis et al. showed that a $frac{2}{3}-mathrm{EFX}$ allocation is guaranteed to exist when number of agents is at most seven. In this paper, we show that a $frac{2}{3}-mathrm{EFX}$ allocation exists for any number of agents when there are at most four distinct valuations. Secondly, we consider a relaxation called $mathrm{EFX}$ with charity, where some goods remain unallocated such that no agent envies the set of unallocated goods. Akrami et al. showed that for $n$ agents and any $varepsilon in left(0, frac{1}{2} ight]$, there exists a $(1-varepsilon)-mathrm{EFX}$ allocation with at most $ ilde{mathcal{O}}((n/varepsilon)^{frac{1}{2}})$ goods to charity. In this paper, we show that a $(1-varepsilon)-mathrm{EFX}$ allocation with a $ ilde{mathcal{O}}(k/varepsilon)^{frac{1}{2}}$ charity exists for any number of agents when there are at most $k$ distinct valuations.

Problem

• Achieving approximate EFX fair allocation for few agent types
• Extending EFX guarantees to four distinct additive valuations
• Reducing charity requirements in EFX with limited valuation types

Innovation

• EFX allocation for four distinct valuations
• EFX with charity for k distinct valuations
• Approximate envy-freeness with bounded charity