A note on uniqueness of electoral equilibrium when the median voter is unobserved

This note considers the unidimensional electoral model in which two candidates seek to maximize their vote totals and do not observe the preferences of the voters. I show that there is a unique equilibrium and, in equilibrium, both candidates locate at the median of the (normalized) sum of distributions of individual voter ideal policies. In fact, this equilibrium is unique among all mixed strategy equilibria.