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. It follows that if the dimension(W)\leqv(𝒟)-2 (where v(𝒟) is the Nakamura number of the voting rule, 𝒟), then the core of 𝒟 cannot be generically empty. For higher dimensions, an "extension" of the voting core, called the heart of 𝒟, 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 𝒟, can be defined and shown to be continuous on the C1-space of smooth convex preference profiles.