Unidimensional Downsian politics: median, utilitarian or what else?

In this note, we show that in the probabilistic version of the standard unidimensional Downsian model with two candidates maximizing their expected pluralities there do not exist Nash equilibria in pure strategies whenever the probabilistic component is arbitrarily small but not degenerate. We argue that empirical research on electoral competition should pay more attention to the magnitude of partisanship in each group of voters, since equilibrium predictions will be very sensitive to that specific electoral parameter