The authors modify the price-setting version of the vertically differentiated duopoly model by Aoki (Effect of Credible Quality Investment with Bertrand and Cournot Competition, 2003) by introducing an extended game in which firms noncooperatively choose the timing of moves at the quality stage. Their results show that there are multiple equilibria in pure strategies, whereby firms always select sequential play at the quality stage. They also investigate the mixed-strategy equilibrium, revealing that the probability of generating out-of-equilibrium outcomes is higher than its complement to one. In the alternative case with full market coverage, the authors show that the quality stage is solved in dominant strategies and therefore the choice of roles becomes irrelevant as the Nash and Stackelberg solutions coincide.