A concise proof of theorem on manipulation of social choice functions

Abstract

Political scientists have for a long time been aware of the fact that most social choice methods used in practice are subject to strategic manipulation in the sense that an individual (or a group of individual), by misrepresenting his preferences, may secure an outcome he prefers to the outcome which would have obtained if he had expressed his sincere preferences. However, when K. Arrow/l/ made the study of social welfare functions a respectable branch of science he deliberately avoided the strategical aspects of voting. It is only recently that the researchers in the area have turned to the possibilities of finding social choice methods which are not subject to strategic manipulation. One of the main results in this direction is a theorem, shown independently by A. Gibbard /3/ and M. Satterthwaite /6/, which deals with social choice functions which, in any voting situation, select a single alternative as the winning alternative. We call such social choice functions resolute social choice functions. The theorem may be formulated as follows: Any resolute social choice function which has at least three possible outcomes is either dictatorial or subject to strategic manipulation by single individuals. The purpose of this paper is to give a comparatively simple proof of this theorem. In Gibbard's proof Arrow's impossibility theorem plays a central role. In fact,