Fair and Efficient Allocation of Indivisible Mixed Manna

TL;DR

This paper introduces a new fairness concept for dividing indivisible mixed items among agents, ensuring fairness and efficiency simultaneously, with algorithms for computing such allocations under various conditions.

Contribution

It establishes the existence of envy-freeness up to $k$ reallocations combined with Pareto efficiency for mixed manna, and provides polynomial-time algorithms for fixed numbers of agents.

Findings

Existence of EFR-(n-1) and Pareto optimal allocations for mixed manna.

Efficient algorithms for computing EFR-(n-1) when the number of agents is fixed.

Tight bounds for EFR allocations when items are goods or chores.

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-${\lfloor n/2 \rfloor}$ allocation exists and can be computed efficiently. Here, the $(n-1)$ bound is tight for chores and $\lfloor n/2 \rfloor$ 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.