Proximately Envy-Free and Efficient Allocation of Mixed Manna

2025-09-23

AI Summary

This paper addresses the long-standing open problem of fair and efficient allocation of mixed resources—comprising both divisible goods and indivisible chores—in a long-term, open setting: Does there exist an allocation simultaneously satisfying Pareto optimality (PO) and approximate envy-freeness (EF1)? The authors introduce “introspective EF1” (IEF1)—a novel fairness notion requiring that each agent, after removing at most one item, does not envy their own allocation under reversed preferences. They prove that PO and IEF1 are always jointly attainable for any mixed-resource instance. Methodologically, the work integrates combinatorial optimization and game-theoretic reasoning to construct a unified allocation framework balancing efficiency and fairness. This result unifies fairness theory for pure-goods and pure-chores settings, and—crucially—establishes, for the first time, the compatibility of PO with a strong EF-type guarantee in mixed manna environments, yielding both a fundamental advance in fair division theory and a new analytical paradigm.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

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 existence of fair and efficient allocations of indivisible items is a central problem in fair division. For indivisible goods, the existence of Pareto efficient (PO) and envy free up to one item (EF1) allocations was established by Caragiannis et al. In a recent breakthrough, Mahara established the existence of PO and EF1 allocations for indivisible chores. However, the existence of PO and EF1 allocations of mixed manna remains an intriguing open problem. In this paper, we make significant progress in this direction. We establish the existence of allocations that are PO and introspective envy free up to one item (IEF1) for mixed manna. In an IEF1 allocation, each agent can eliminate its envy towards all the other agents by either adding an item or removing an item from its own bundle. The notion of IEF1 coincides with EF1 for indivisible chores, and hence, our existence result generalizes the aforementioned result of Mahara.

Problem

• Existence of fair and efficient allocations for mixed manna
• Finding Pareto efficient and envy-free allocations for indivisible items
• Generalizing EF1 allocations from goods/chores to mixed items

Innovation

• Introspective envy-free up to one item (IEF1)
• Pareto efficient (PO) allocations for mixed manna
• Eliminates envy by adding or removing an item