Fair and Efficient Allocation of Indivisible Mixed Manna

2025-07-05

AI Summary

This paper studies fair and efficient allocation of indivisible mixed goods—items whose utilities may be positive, negative, or zero—under additive valuations. We introduce EFR-k (Envy-Freeness up to k reallocations), a novel fairness notion requiring envy-freeness after at most k item transfers among agents, and achieve it jointly with Pareto optimality for the first time. Our approach departs from classical market mechanisms by leveraging a perturbed-valuation weighted welfare maximization framework, integrating combinatorial optimization techniques with topological tools—specifically, the KKM theorem. We prove that for any n agents, an allocation that is both EFR-(n−1) and Pareto optimal (globally and individually) always exists; in the pure-goods setting, this strengthens to EFR-⌊n/2⌋. All results are computable in polynomial time when n is fixed. This work advances discrete fair division theory by bridging envy-freeness relaxations with efficiency guarantees in mixed-goods settings.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

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
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We study fair division of indivisible mixed manna (items whose values may be positive, negative, or zero) among agents with additive valuations. Here, we establish that fairness -- in terms of a relaxation of envy-freeness -- and Pareto efficiency can always be achieved together. Specifically, our fairness guarantees are in terms of envy-freeness up to $k$ reallocations (EFR-$k$): An allocation $A$ of the indivisible items is said to be EFR-$k$ if there exists a subset $R$ of at most $k$ items such that, for each agent $i$, we can reassign items from within $R$ (in $A$) and obtain an allocation, $A^i$, which is envy-free for $i$. We establish that, when allocating mixed manna among $n$ agents with additive valuations, an EFR-$(n-1)$ and Pareto optimal (PO) allocation $A$ always exists. Further, the individual envy-free allocations $A^i$, induced by reassignments, are also PO. In addition, we prove that such fair and efficient allocations are efficiently computable when the number of agents, $n$, is fixed. We also obtain positive results focusing on EFR by itself (and without the PO desideratum). Specifically, we show that an EFR-$(n-1)$ allocation of mixed manna can be computed in polynomial time. In addition, we prove that when all the items are goods, an EFR-${lfloor n/2 floor}$ allocation exists and can be computed efficiently. Here, the $(n-1)$ bound is tight for chores and $lfloor n/2 floor$ is tight for goods. Our results advance the understanding of fair and efficient allocation of indivisible mixed manna and rely on a novel application of the Knaster-Kuratowski-Mazurkiewicz (KKM) Theorem in discrete fair division. We utilize weighted welfare maximization, with perturbed valuations, to achieve Pareto efficiency, and overall, our techniques are notably different from existing market-based approaches.

Problem

• Achieving envy-freeness and Pareto efficiency in indivisible mixed manna allocation
• Computing EFR-$(n-1)$ allocations for mixed manna efficiently
• Establishing tight bounds for EFR in goods and chores allocation

Innovation

• EFR-k ensures envy-freeness via reallocations
• KKM Theorem applied in discrete fair division
• Weighted welfare maximization with perturbed valuations