Fair and Efficient Balanced Allocations for Additive Valuations

2026-08-06

AI Summary

This study addresses the fair and efficient allocation of indivisible goods under an equitability constraint—requiring that the number of items received by any two agents differs by at most one. For agents with additive valuations, it establishes the first existence result of an equitable allocation that simultaneously satisfies envy-freeness up to one good (EF1) and fractional Pareto optimality (fPO), thereby extending beyond prior results limited to binary or bivalued valuations. The proof combines a weighted welfare duality framework with the KKM lemma and introduces a novel price-interleaving lemma to overcome key technical barriers. The result is further generalized to partition matroid (categorical) constraints, where a relaxed EF1 guarantee holds: envy can be eliminated by removing at most one item from each category.

Abstract

We study the existence of fair and efficient allocations of indivisible goods under the balancedness constraint, which requires that any two agents' bundles differ in size by at most one. Our main result establishes the existence of balanced allocations that satisfy envy-freeness up to one good (EF1) and fractional Pareto optimality (fPO) for arbitrary additive valuations. This generalizes a recent result of Kawase et al. (2026), which establishes existence only for personalized bivalued valuations or when there are at most two distinct valuation types. Our proof applies the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma to a weighted-welfare duality framework and develops a novel price-interlacing lemma to overcome barriers encountered by prior work. We extend this technique to category constraints, also known as partition-matroid constraints. In this setting, we establish the existence of an fPO allocation satisfying a weaker, category-sensitive relaxation of EF1, under which envy can be eliminated by removing at most one good from each category. All proofs in this paper were obtained using GPT-5.6-Sol with guidance from the authors. The authors verified the proofs, expanded the exposition, and simplified the arguments with assistance from GPT-5.6-Sol and Claude Fable 5.

Problem

• balanced allocations
• envy-freeness up to one good
• fractional Pareto optimality
• additive valuations
• indivisible goods

Innovation

• balanced allocation
• EF1
• fractional Pareto optimality
• KKM lemma
• price-interlacing lemma