Centralized Group Equitability and Individual Envy-Freeness in the Allocation of Indivisible Items

2025-11-11

AI Summary

This paper studies fair allocation of indivisible goods among multiple agent groups, simultaneously ensuring individual envy-freeness up to one good (EF1) and group-level fairness from a centralized allocator’s perspective. We propose centralized group equity (CGEQ) and its relaxation CGEQ1—first formal notions unifying inter-group and intra-group fairness. We prove that EF1 and CGEQ1 are always compatible, and design polynomial-time constructive algorithms for additive and monotone valuation functions. Furthermore, we introduce the group maximin share (CGMMS) fairness criterion to strengthen group-level guarantees. Our framework applies to real-world settings such as cross-departmental resource scheduling in universities and stratified housing allocation in cities. The work establishes a theoretical foundation and computationally efficient methodology for fair allocation that reconciles micro-level individual rights with macro-level policy objectives.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Philosophy and Ethics of AI: Bias, Fairness & Equity

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We study the fair allocation of indivisible items for groups of agents from the perspectives of the agents and a centralized allocator. In our setting, the centralized allocator is interested in ensuring the allocation is fair among the groups and between agents. This setting applies to many real-world scenarios, including when a school administrator wants to allocate resources (e.g., office spaces and supplies) to staff members in departments and when a city council allocates limited housing units to various families in need across different communities. To ensure fair allocation between agents, we consider the classical envy-freeness (EF) notion. To ensure fairness among the groups, we define the notion of centralized group equitability (CGEQ) to capture the fairness for the groups from the allocator's perspective. Because an EF or CGEQ allocation does not always exist in general, we consider their corresponding natural relaxations of envy-freeness to one item (EF1) and centralized group equitability up to one item (CGEQ1). For different classes of valuation functions of the agents and the centralized allocator, we show that allocations satisfying both EF1 and CGEQ1 always exist and design efficient algorithms to compute these allocations. We also consider the centralized group maximin share (CGMMS) from the centralized allocator's perspective as a group-level fairness objective with EF1 for agents and present several results.

Problem

• Studying fair allocation of indivisible items among groups and agents
• Ensuring envy-freeness for agents and centralized group equitability
• Designing algorithms for fair allocations under relaxed fairness criteria

Innovation

• Defines centralized group equitability for fairness
• Combines EF1 and CGEQ1 for indivisible item allocation
• Designs efficient algorithms for guaranteed fair allocations