The C1 topology on the space of smooth preference profiles |
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Authors: | Norman Schofield |
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Institution: | (1) Center in Political Economy, Campus Box 1208, Washington University, One Brooking Drive, St. Louis, MO 63130-4899, USA (e-mail: schofld@wuecon.wustl.edu), US |
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Abstract: | This paper defines a fine C
1-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)≤v(?)−2 (where v(?) is the Nakamura number of the voting rule, ?), then the core of ? cannot be generically empty. For higher dimensions,
an “extension” of the voting core, called the heart of ?, is proposed. The heart is a generalization of the “uncovered set”.
It is shown to be non-empty and closed in general. On the C
1-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 ?, can be defined and shown to be continuous on the C
1-space of smooth convex preference profiles.
Received: 3 April 1995/Accepted: 8 April 1998 |
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Keywords: | |
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