Fair Allocation with Initial Utilities

2026-02-16

AI Summary

This work addresses the limitations of traditional fairness criteria—such as envy-freeness up to one good (EF1)—in the allocation of indivisible resources, which often ignore agents’ initial utility disparities and may consequently yield no feasible solution or exacerbate outcome inequality. To overcome this, the paper introduces a novel fairness notion, minimum-EF1-init, which remains always satisfiable and explicitly accounts for pre-existing utility differences while promoting equitable outcomes. Focusing on additive valuations with initial endowments, the authors demonstrate that EF1 allocations may not exist and are computationally intractable in this setting. In contrast, they propose an extended round-robin algorithm that efficiently computes a complete allocation satisfying minimum-EF1-init in polynomial time, thereby offering a theoretically grounded and practically viable approach to fair division under initial inequality.

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

The problem of allocating indivisible resources to agents arises in a wide range of domains, including treatment distribution and social support programs. An important goal in algorithm design for this problem is fairness, where the focus in previous work has been on ensuring that the computed allocation provides equal treatment to everyone. However, this perspective disregards that agents may start from unequal initial positions, which is crucial to consider in settings where fairness is understood as equality of outcome. In such settings, the goal is to create an equal final outcome for everyone by leveling initial inequalities through the allocated resources. To close this gap, focusing on agents with additive utilities, we extend the classic model by assigning each agent an initial utility and study the existence and computational complexity of several new fairness notions following the principle of equality of outcome. Among others, we show that complete allocations satisfying a direct analog of envy-freeness up to one resource (EF1) may fail to exist and are computationally hard to find, forming a contrast to the classic setting without initial utilities. We propose a new, always satisfiable fairness notion, called minimum-EF1-init and design a polynomial-time algorithm based on an extended round-robin procedure to compute complete allocations satisfying this notion.

Problem

• fair allocation
• initial utilities
• equality of outcome
• indivisible resources
• envy-freeness

Innovation

• initial utilities
• equality of outcome
• minimum-EF1-init
• indivisible resource allocation
• fair division