AI 中文总结
该研究针对加性估值，证明了满足EF1与fPO的均衡分配存在，推广了Kawase等人的结果，还将技术扩展到类别约束场景，相关证明借助GPT-5.6-Sol完成。
AI 中文摘要
我们研究了在均衡性约束下不可分割商品的公平且高效分配的存在性，该约束要求任意两个智能体的商品束大小相差不超过1。我们的主要结果证明，对于任意加性估值，存在满足“至多一件商品的妒恨规避（EF1）”和“分数帕累托最优（fPO）”的均衡分配。这推广了Kawase等人（2026）的最新结果，该结果仅证明了个性化双值估值或至多两种不同估值类型下的存在性。我们的证明将Knaster-Kuratowski-Mazurkiewicz（KKM）引理应用于加权福利对偶框架，并提出了一种新颖的价格交错引理以克服先前工作遇到的障碍。我们将该技术扩展到类别约束（也称为划分拟阵约束），在该设定下，我们证明了存在满足fPO且满足更弱的、类别敏感的EF1松弛的分配，该松弛下可通过从每个类别中移除至多一件商品来消除妒恨。本文所有证明均使用GPT-5.6-Sol完成，作者在其协助下验证了证明、扩展了阐述并简化了论证，同时也得到了Claude Fable 5的协助。
英文摘要
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.
