Decomposability and Strategy-proofness in Multidimensional Models

We introduce the notion of a multidimensional hybrid preference domain on a (finite) set of alternatives that is a Cartesian product of finitely many components. We study strategy-proof rules on this domain and show that every such rule can be decomposed into component-wise strategy proof rules. More importantly, we show that under a suitable ``richness'' condition, every domain of preferences that reconciles decomposability of rules with strategy-proofness must be a multidimensional hybrid domain. We finally identify an intuitive weakening of separability that explains how a multidimensional hybrid domain may arise in a public goods provision model.