The C1 topology on the space of smooth preference profiles

This paper defines a fine C1-topology for smooth preferences on a "policy space", W, and shows that the set of convex preference profiles contains open sets in this topology. <p> It follows that if the dimension(W)\leqv(&Dscr;)-2 (where v(&Dscr;) is the Nakamura number of the voting rule, &Dscr;), then the core of &Dscr; cannot be generically empty. For higher dimensions, an "extension" of the voting core, called the heart of &Dscr;, is proposed. The heart is a generalization of the "uncovered set". It is shown to be non-empty and closed in general. On the C1-space of convex preference profiles, the heart is Paretian. Moreover, the heart correspondence is lower hemi-continuous and admits a continuous selection. Thus the heart converges to the core when the latter exists. Using this, an aggregator, compatible with &Dscr;, can be defined and shown to be continuous on the C1-space of smooth convex preference profiles.