Two statements of the Duggan-Schwartz theorem

TL;DR

This paper clarifies the relationship between two formulations of the Duggan-Schwartz theorem, demonstrating their equivalence and shedding light on the foundational aspects of strategy-proof social choice correspondences.

Contribution

It establishes the equivalence between the original Duggan-Schwartz theorem and Taylor's reformulation, clarifying their conceptual relationship.

Findings

Proves the equivalence of the two statements of the Duggan-Schwartz theorem

Clarifies the foundational understanding of strategy-proof social choice

Highlights the historical attribution of the theorem's formulation

Abstract

The Duggan-Schwartz theorem (Duggan and Schwartz, 1992) is a famous result concerning strategy-proof social choice correspondences, often stated as "A social choice correspondence that can be manipulated by neither an optimist nor a pessimist has a weak dictator". However, this formulation is actually due to Taylor (2002), and the original theorem, at face value, looks rather different. In this note we show that the two are in fact equivalent.