Implementation in multidimensional domains with ordinal restrictions

We consider implementation of a deterministic allocation rule using transfers in quasi-linear private values environments. We show that if the type space is a multidimensional domain satisfying some ordinal restrictions, then an allocation rule is implementable in such a domain if and only if it satisfies a familiar and simple condition called 2-cycle monotonicity. Our ordinal restrictions cover type spaces which are non-convex, e.g., the single peaked domain and its generalizations. We apply our result to show that in the single peaked domain, a local version of 2-cycle monotonicity is necessary and sufficient for implementation and every locally incentive compatible mechanism is incentive compatible.