We reinvestigate a theoretical result by Rhim and Cooper (2005), who provide a uniqueness condition for price equilibria in a two-stage competitive product positioning model. We show that this condition is very restrictive by providing a simple proof for the fact that it eventually results in parametric pricing. As a consequence, it misses the two-stage characteristic that the model originally aims at. (C) 2015 Elsevier B.V. All rights reserved.