Generalized medians and electoral competition with valence

I establish conditions for existence of pure strategy equilibria in K-candidate Downsian electoral competition (K ≥ 2) with valence when the voting rule is monotonic, generalizing existing results to non-proper rules and possibly continuous electorates. The conditions are sufficient when K ≥ 2 and (essentially) necessary in the K = 2 candidate case. They compare the size of one candidate's valence advantage to the radius of a generalized median pivotal ball (P-ball). I flesh out the difference of this generalized median with a recent alternative which, in turn, I characterize both on the basis of a weaker median property and using pivotal hyperplanes.