On the nature of equilibria in a Downsian model with candidate valence

I analyze mixed strategy equilibria in a Downsian model with two office-motivated candidates in which one candidate is endowed with a sufficiently large valence advantage that a voter might prefer this candidate even if the voter strictly prefers the other candidate's policies. There is a discrete one-dimensional policy space and the preferences of the median voter are uncertain. I show that there is a range of moderate policies with no gaps that are optimal for the advantaged candidate. There is also a range of liberal policies with no gaps and a corresponding range of conservative policies with no gaps that are optimal actions for the disadvantaged candidate. The upper and lower bounds on these ranges of policies vary in predictable ways with the size of the advantaged candidate's advantage.