Fair Allocation with Binary Valuations for Mixed Divisible and Indivisible Goods

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.