The Hotelling–Downs model of spatial competition is used to investigate the strategic position-taking behavior of firms or political parties under scoring rules. Previous studies of non-convergent Nash equilibria – equilibria in which divergent positions are chosen – found that they often do not exist and, when they do, they are fairly symmetric. In particular, this is true for convex scoring rules (Cahan and Slinko, 2017). Here, we investigate non-convergent equilibria for the broad class of weakly concave scoring rules. Surprisingly, we find that only asymmetric equilibria can exist, and we present several examples.