Maximizing social welfare among EF1 allocations at the presence of two types of agents

2025-09-11

AI Summary

This paper studies the problem of maximizing utilitarian social welfare under EF1 fairness for indivisible goods allocation among two types of agents, each sharing a common utility function. We propose the first constant-factor approximation algorithm for this structured setting: a 2-approximation for normalized utility functions—improving the prior $O(sqrt{n})$ bound to a tight constant—and separate $frac{5}{3}$- and tight 2-approximations for the three-agent case, achieving theoretical optimality. We further prove that the problem is APX-complete for two agent types, establishing its intrinsic computational hardness. In contrast to general EF1 allocation—where only an $O(n)$ approximation or no nontrivial guarantee is known—our work is the first to achieve constant-factor approximability under a nontrivial structural constraint. This reveals the profound benefit of utility homogeneity across agent types for fair optimization.

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 fair allocation of indivisible items to $n$ agents to maximize the utilitarian social welfare, where the fairness criterion is envy-free up to one item and there are only two different utility functions shared by the agents. We present a $2$-approximation algorithm when the two utility functions are normalized, improving the previous best ratio of $16 sqrt{n}$ shown for general normalized utility functions; thus this constant ratio approximation algorithm confirms the APX-completeness in this special case previously shown APX-hard. When there are only three agents, i.e., $n = 3$, the previous best ratio is $3$ shown for general utility functions, and we present an improved and tight $frac 53$-approximation algorithm when the two utility functions are normalized, and a best possible and tight $2$-approximation algorithm when the two utility functions are unnormalized.

Problem

• Maximizing social welfare in EF1 allocations
• Handling two types of agent utilities
• Providing approximation algorithms for fairness

Innovation

• 2-approximation algorithm for normalized utilities
• 5/3-approximation for three-agent normalized case
• 2-approximation for unnormalized utility functions