JOURNAL OF ECONOMIC THEORY 42, 382-386 (1987) On Constant Maskin Monotonic Social Choice Functions* TATSUYOSHI SAIJO Department of’ Economics, University of Cahyornia. Santa Barbara, California 93106 Received December 1 I, 1985; revised August 15, 1986 A social choice function satisfying a “dual dominance” condition is Maskin monotonic if and only if it is constant. Hence, any non-constant social choice function satisfying dual dominance is not Nash implementable. Among the impor- tant examples are all social choice functions on unrestricted domains. On the other hand, non-constant social choice functions such as a Walrasian performance function in economic environments need not satisfy dual dominance. Journal of Economic Literature Classification Numbers: 024 and 025. ( 1987 Academic Press. Inc. 1. INTRODUCTION We shall show that the Maskin monotonicity (MM) of a social choice function (s.c.f.) (see [3]) is equivalent to constancy of that s.c.f., provided it satisfies what we call the dual dominance (DD) condition. Any s.c.f. which is defined on all possible preference profiles, i.e., on the unrestricted domain, satisfies DD. Therefore, any MM s.c.f. on an unrestricted domain is constant. No constant s.c.f. on an unrestricted domain is dictatorial. We will examine two well-known results on dictatorial s.c.f.s. in light of this constancy theorem. The first implication of our result is that the Dasgupta-Ham- *This note is based upon Chapter 3 of the author’s dissertation [S] presented to the Graduate School of the University of Minnesota. The author wishes to thank Professors Leonid Hurwicz, Takatoshi Ito, Donald Lien, Richard McKelvey, and Stephen Turnbull for inspiring comments and suggestions. The author especially thanks an associate editor of this journal and anonymous referees for comments that greatly improve this note. The respon- sibility for any errors remains with the author. Financial support from a doctoral fellowship from the Graduate School of the University of Minnesota is gratefully acknowledged. The author is also grateful to the Division of the Humanities and Social Sciences of the California Institute of Technology for providing an excellent research environment. 382 0022-0531/87 $3.00 Copyright Q 1987 by Academic Press, Inc. All rights of reproduction m any form reserved