Robertsʼ Theorem with neutrality: A social welfare ordering approach

We consider dominant strategy implementation in private values settings, when agents have multi-dimensional types, the set of alternatives is finite, monetary transfers are allowed, and agents have quasi-linear utilities. We focus on private-value environments. We show that any implementable and neutral social choice function must be a weighted welfare maximizer if the type space of every agent is an m-dimensional open interval, where m is the number of alternatives. When the type space of every agent is unrestricted, Robertsʼ Theorem with neutrality (Roberts, 1979) becomes a corollary to our result. Our proof technique uses a social welfare ordering approach, commonly used in aggregation literature in social choice theory. We also prove the general (affine maximizer) version of Robertsʼ Theorem for unrestricted type spaces of agents using this approach.