Lemma 7.1. Assume the conditions on the voter distribution detailed for our illustrative example. Then, under the unified turnout model, given b > TA, b > TI, and b < ∞, any possible equilibrium configuration (D, R) must satisfy D < μ < R.

Proof. Note first that given the voter distribution assumed for our illustrative example, a candidate obtains as many votes as his opponent whenever he locates at a position symmetric to his opponent's position with respect to μ, the position of the median voter. This implies that both candidates must receive equal vote shares at equilibrium. Now suppose that R < μ. Then if D pairs with R, each candidate receives support only from his own partisans and only from those in a common interval that is symmetric about the common point, D = R < μ(note that given b > TA, b > TI, and b < ∞, this interval must be of finite, nonzero length). Because this point is closer to the median Democratic voter than to the median Republican, this interval contains more Democrats than Republicans. Hence when R < μ, the Democratic candidate can defeat the Republican, which violates the condition that any equilibrium configuration must find the candidates receiving equal vote shares.

Next consider the case where R = μ. Now the Democratic candidate can defeat the Republican by locating at any point in the policy interval [μD, μ] such that b > (R − D).