Two Proofs of the Gibbard-Satterthwaite Theorem on the Possibility of a Strategy-Proof Social Choice Function

Abstract

We present two proofs of a result which was formulated independently by A. Gibbard [2] and M. Satterthwaite [3]. Their theorem provides an attractive new way of viewing Arrow’s classic result on Social Welfare Functions [1]. By requiring that strategic considerations cannot be beneficially employed, it frees the statement of the ‘General Possibility Theorem’ from the assumption of independence of irrelevant alternatives. The theorem demonstrates that all mechanisms for passing from individual preferences (over three or more alternatives) to a social choice are vulnerable to strategic play.

We acknowledge the helpful comments of Allan Gibbard, Mark Satterthwaite, and Leonid Hurwicz. David Schmeidler’s research was partially supported by a grant from the Foerder Institute at Tel-Aviv University and Hugo Sonnenschein’s by a grant from the National Science Foundation of the United States. Our first proof follows Gibbard’s [2] in its reliance on Arrow’s General Possibility Theorem, and Satterthwaite’s [3] in that the case of strongly ordered preference is taken up first. The extension from strongly to weakly ordered preferences is new. The second proof is entirely new.