A characterization of the rational mean neat voting rules

A mean proximity rule is a voting rule having a mean proximity representation in Euclidean space. Legal ballots are represented as vectors that form the representing polytope. An output plot function determines a location for each possible election output in the same space, and these locations decompose the polytope into proximity regions according to which output is closest. The election outcome is then determined by which region(s) contain the mean position of all ballots cast. Mean neat rules are obtained by relaxing the requirement that the regions be determined by proximity, insisting only that they be neatly separable by a hyperplane. If each of these hyperplanes contains a dense set of rational points (vectors with all rational components), the mean neat voting rule is said to be rational. The aim of this article is to prove that consistency and connectedness are necessary and sufficient conditions for mean neat rationality of any voting rule that is anonymous. Connectedness can be viewed as a strong form of continuity, with an intuitive content related to the Intermediate Value Theorem (or to a discrete analogue of this theorem). The proof relies on a recent result in convexity theory [D. Cervone, W.S. Zwicker, Convex decompositions, J. Convex Anal. 2008 (in press)] and suggests a conjecture: if we relax connectedness to continuity, the class so characterized is that of the mean neat voting rules. This latter class properly contains all intuitive scoring rules.