Characterizing implementable allocation rules in multi-dimensional environments

We study characterizations of implementable allocation rules when types are multi-dimensional, monetary transfers are allowed, and agents have quasi-linear preferences over outcomes and transfers. Every outcome is associated with a valuation function that maps an agent's type to his value for this outcome. The set of types are assumed to be convex. Our main characterization theorem shows that allocation rules are implementable if and only if they are implementable on any two-dimensional convex subset of the type set. For finite sets of outcomes and continuous valuation functions, they are implementable if and only if they are implementable on every one-dimensional subset of the type set. This extends a characterization result by Saks and Yu (Weak monotonicity suffices for truthfulness on convex domains, pp 286-293, 2005) from models with linear valuation functions to arbitrary continuous valuation functions, and provides a simple proof of their result. Modeling multi-dimensional mechanism design the way we propose it here is of relevance whenever types are given by few parameters, while the set of possible outcomes is large, and when values for outcomes are non-linear functions in types.