Beyond Cake Cutting: Allocating Homogeneous Divisible Goods

The problem of fair division known as "cake cutting" has been the focus of multiple papers spanning several decades. The most prominent problem in this line of work has been to bound the query complexity of computing an envy-free outcome in the Robertson-Webb query model. However, the root of this problem's complexity is somewhat artificial: the agents' values are assumed to be additive across different pieces of the "cake" but infinitely complicated within each piece. This is unrealistic in most of the motivating examples, where the cake represents a finite collection of homogeneous goods. We address this issue by introducing a fair division model that more accurately captures these applications: the value that an agent gains from a given good depends only on the amount of the good they receive, yet it can be an arbitrary function of this amount, allowing the agents to express preferences that go beyond standard cake cutting. In this model, we study the query complexity of computing allocations that are not just envy-free, but also approximately Pareto optimal among all envy-free allocations. Using a novel flow-based approach, we show that we can encode the ex-post feasibility of randomized allocations via a polynomial number of constraints, which reduces our problem to solving a linear program.