Abstract
We characterize additive representations on subsets of product spaces with an empty interior such as simplexes and certain homeomorphisms thereof. Previously, all additive representation theorems only applied to spaces in which any coordinate can be changed without changing any of the other coordinates. We identify a novel preference condition that is necessary and sufficient for the existence of additive representations. Our results provide, for instance, a characterization of utilitarianism on the Pareto frontier of a cake division problem.
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