The fair allocation of mixed goods, consisting of both divisible and indivisible goods, among agents with heterogeneous preferences, has been a prominent topic of study in economics and computer science. In this paper, we investigate the nature of fair allocations when agents have binary valuations. We define an allocation as fair if its utility vector minimizes a symmetric strictly convex function, which includes conventional fairness criteria such as maximum egalitarian social welfare and maximum Nash social welfare. While a good structure is known for the continuous case (where only divisible goods exist) or the discrete case (where only indivisible goods exist), deriving such a structure in the hybrid case remains challenging. Our contributions are twofold. First, we demonstrate that the hybrid case does not inherit some of the nice properties of continuous or discrete cases, while it does inherit the proximity theorem. Second, we analyze the computational complexity of finding a fair allocation of mixed goods based on the proximity theorem. In particular, we provide a polynomial-time algorithm for the case when all divisible goods are identical and homogeneous, and demonstrate that the problem is NP-hard in general. Our results also contribute to a deeper understanding of the hybrid convex analysis.

翻译：混合商品（包含可分割与不可分割商品）在具有异质性偏好的代理人之间的公平分配，一直是经济学与计算机科学领域的重要研究课题。本文研究了代理人对商品具有二元估值时公平分配的性质。我们将公平分配定义为效用向量能最小化一个对称严格凸函数的情形，该函数涵盖最大平等主义社会福利和最大纳什社会福利等传统公平准则。尽管连续情形（仅存在可分割商品）或离散情形（仅存在不可分割商品）中已知良好的结构性质，但混合情形下推导此类结构仍具挑战性。我们的贡献体现在两方面：首先，证明混合情形虽不继承连续或离散情形中的某些优良性质，但确实继承邻近定理；其次，基于邻近定理分析混合商品公平分配的计算复杂性。特别地，我们针对所有可分割商品均为同质同构的情形提出多项式时间算法，并证明该问题在一般情况下是NP难的。本研究结果亦有助于深化对混合凸分析的理解。