Best-response potential for Hotelling pure location games

We revisit two-person one-dimensional pure location games à la Anderson et al. (1992) and show that they admit continuous best-response potential functions (Voorneveld, 2000) if demand is sufficiently elastic (to the extent that the Principle of Minimum Differentiation fails); if demand is not that elastic (or is completely inelastic) they still admit continuous quasi-potential functions (Schipper, 2004). We also show that, even if a continuous best-response potential function exists, a generalized ordinal potential function (Monderer and Shapley, 1996) need not exist.