On the structure of division rules

We consider the problem of dividing one unit of an infinitely divisible object among a finite number of agents. We provide a characterization of all single-peaked domains on which the uniform rule is the unique division rule satisfying efficiency, strategy-proofness, and equal treatment of equals (ETE). We also provide a class of division rules satisfying these properties on the remaining single-peaked domains. Next, we consider non single-peaked domains and provide a characterization of all such domains on which the uniform rule satisfies efficiency, strategy-proofness, and ETE. We also show that under some mild richness conditions the uniform rule is the unique rule satisfying the mentioned properties on these domains. Finally, we provide a class of division rules satisfying efficiency, strategy-proofness, and ETE on the remaining non single-peaked domains. We conclude the paper by providing a wide range of applications to justify the usefulness of our results.