Roberts' Theorem with Neutrality: A Social Welfare Ordering Approach

TL;DR

This paper extends Roberts' theorem to multi-dimensional types with neutrality, showing that implementable and neutral social choice functions are weighted welfare maximizers, using a social welfare ordering approach.

Contribution

It introduces a social welfare ordering method to prove Roberts' theorem with neutrality for multi-dimensional type spaces, generalizing previous results.

Findings

Implementable and neutral social choice functions are weighted welfare maximizers.

The proof technique uses social welfare orderings from social choice theory.

The paper generalizes Roberts' theorem to unrestricted type spaces.

Abstract

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 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, Roberts' theorem with neutrality \cite{Roberts79} becomes a corollary to our result. Our proof technique uses a {\em social welfare ordering} approach, commonly used in aggregation literature in social choice theory. We also prove the general (affine maximizer) version of Roberts' theorem for unrestricted type spaces of agents using this approach.