🤖 AI Summary
This study addresses the existence of random allocations that are both weakly Pareto efficient and envy-free under a general setting encompassing indivisible goods, divisible resources, and complex scenarios involving temporal or heterogeneous commodities. By constructing a unified framework, allocations are modeled as probability measures on a compact metric space, with agents’ preferences represented by continuous, concave utility functions defined over the space of probability measures. Leveraging tools from functional analysis, measure theory, and convex analysis, the paper establishes—for the first time—the existence of such allocations in settings that include non-atomless preferences, and shows they can be represented as probability measures supported on finite partitions. This framework not only unifies classical problems such as school choice and fair cake-cutting but also extends to novel applications beyond the scope of existing theory.
📝 Abstract
I provide a unified framework to establish the existence of a weak Pareto efficient, envy-free allocation in general settings: random allocations are probability measures on a compact metric space, and preferences of agents are represented by continuous, concave utility function on the space of probability measures. The generality of my setting nests the existence results for small spaces with indivisibles -- the list of prominent applications includes the school assignment problem and the house allocation problem. The technique developed to prove the existence also applies to allocation problems with divisibles, like fair cake-cutting or land-division problems. Here I also show that even when agents' preferences are not atomless, the allocation in question can be represented as a probability measure over partitions with finite support. Last but not least, I apply the existence result to new allocation problems that no existing framework encompasses. These include allocation of indivisible goods or services over time and allocation of differentiated goods.