The Downsian model predicts divergence

The 'Downsian' model is widely held to predict the convergence of parties to the median in two-party systems with unidimensional policy spaces. We show this is not true, if voters are uncertain. This paper gives an exposition of the two-party divergence results, including (1) if both candidates are free to move, but the incumbent has lower variance of policy action as perceived by voters, then there is no equilibrium, even if preferences are 'single-peaked'; (2) if the lower-variance candidate is forced to run on a fixed record, then an equilibrium platform exists for the challenger, but it always diverges from incumbent's platform; (3) if the incumbent's perceived policy once in office has a lower variance than any prospective challenger, the incumbent may win each of a long series of elections even if the incumbent's policy choices diverge from the underlying median of the distribution of voter preferences.