AI 中文总结
该研究针对三智能体在一般赋值及层状约束下的公平分配问题，证明了均衡EF$1^c_g$分配的存在性，指出无单调性时无法保证均衡EF1分配，且借助GPT-5.6完成并验证了相关证明。
AI 中文摘要
我们研究一般集合赋值下不可分物品的公平分配问题。我们证明，对于三智能体及任意实值赋值的所有实例，均存在一种均衡分配，满足无嫉妒至多一件物品和一项杂务（EF$1^c_g$）。当每个赋值为非递减或非递增时，这直接意味着均衡EF1分配。我们还表明，若无单调性，无法保证均衡EF1分配：存在一个包含三智能体、九件物品且具有相同非单调赋值的实例，不存在均衡EF1分配。随后我们考虑常见的层状拟阵约束，只要存在完整可行分配，我们证明对于任意赋值，存在一种完整可行分配，满足均衡且无嫉妒至多两件物品和两项杂务（EF$2^c_g$）。该分配还可满足：分配给三智能体的每个层状集合中的物品数量差值至多为2。本文多数证明使用GPT-5.6获得，随后我们借助GPT-5.6验证了证明的正确性并完善了其表述与论证。
英文摘要
We study fair allocations of indivisible items under general set valuations. We prove that every instance with three agents and arbitrary real-valued valuations admits a balanced allocation that is envy-free up to one good and one chore (EF$1^c_g$). This directly implies balanced EF$1$ when each valuation is either monotone nondecreasing or monotone nonincreasing. We also show that balanced EF$1$ cannot be guaranteed without monotonicity: there exists an instance with three agents, nine items, and identical nonmonotone valuations that admits no balanced EF$1$ allocation. We then consider a common laminar matroid constraint. Whenever a complete feasible allocation exists, we prove that there is a complete feasible allocation that is balanced and envy-free up to two goods and two chores (EF$2^c_g$) for arbitrary valuations. The allocation can additionally be chosen so that the numbers of items from every laminar set assigned to the three agents differ by at most two. Most proofs in this paper were obtained using GPT-5.6. We subsequently verified the proofs for correctness and refined their exposition and arguments, also with the aid of GPT-5.6.
