Straightforward Elections, Unanimity, and Phantom Voters

Abstract

Nonmanipulable direct revelation social choice functions are characterized for societies where the space of alternatives is a euclidean space and all voters have separable preferences with a global optimum. If a nonmanipulable choice function satisfies a weak unanimity-respecting condition (which is equivalent to having an unrestricted range) then it will depend only on voters' ideal points. Further, such a choice function will decompose into a product of one-dimensional mechanisms in the sense that each coordinate of the chosen point depends only on the respective coordinate of the voter's ideal points. Each coordinate function will also be nonmanipulable and respect unanimity. Such one-dimensional mechanisms are uncompromising in the sense that voters cannot take an extreme position to influence the choice to their advantage. Two characterizations of uncompromising choice functions are presented. One is in terms of a continuity condition, the other in terms of "phantom voters," i.e., those points which are chosen which are not any voter's ideal point. There are many such mechanisms which are not dictatorial. However, if differentiability is required of the choice function, this forces it to be either constant or dictatorial. In the multidimensional case, nonseparability of preferences leads to dictatorship, even if preferences are restricted to be quadratic.

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