Fair and Efficient Allocation of Indivisible Mixed Manna
Siddharth Barman, Vishwa Prakash HV, Aditi Sethia, Mashbat Suzuki
TL;DR
This work addresses fair division of indivisible mixed manna among $n$ agents with additive valuations, aiming to achieve fairness and efficiency simultaneously. It introduces the envy-freeness up to $k$ reallocations (EFR-$k$) notion and proves that an $ ext{EFR-}(n-1)$ and Pareto-optimal allocation always exists for mixed manna, with the corresponding envy-free reassignments $ ext{A}^i$ also PO; the construction hinges on a novel application of the Knaster–Kuratowski–Mazurkiewicz (KKM) theorem to a perturbed, weighted welfare framework. The authors also show that, for a fixed number of agents, such allocations can be computed in polynomial time, and they obtain targeted results: an $ ext{EFR-}(n-1)$ allocation for mixed manna, an $ ext{EFR-}igl floor n/2 igr floor$ allocation for goods, and tightness results for the bounds. An important part of the work is the separation of duties between goods and chores, as well as hardness results for deciding whether a given allocation is $ ext{EFR-}k$. The approach offers a distinct, topological path to fairness that complements market-based methods and advances understanding of discrete fair division with mixed valuations.
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.