  We use Hotelling's spatial model of competition to investigate the
position-taking behaviour of political candidates under a class of electoral
systems known as scoring rules. In a scoring rule election, voters rank all the
candidates running for office, following which the candidates are assigned
points according to a vector of nonincreasing scores. Convergent Nash
equilibria in which all candidates adopt the same policy were characterised by
Cox (1987). Here, we investigate nonconvergent equilibria, where candidates
adopt divergent policies. We identify a number of classes of scoring rules
exhibiting a range of different equilibrium properties. For some of these,
nonconvergent equilibria do not exist. For others, nonconvergent equilibria in
which candidates cluster at positions spread across the issue space are
observed. In particular, we prove that the class of convex rules does not have
Nash equilibria (convergent or nonconvergent) with the exception of some
derivatives of Borda rule. Finally, we examine the special cases of four-,
five- and six- candidate elections. In the former two cases, we provide a
complete characterisation of nonconvergent equilibria.
