In this paper we investigate a model of duopolistic competition in an uncertain environment where the attitudes of the firms towards uncertainty are incorporated. In particular, we analyze an extension of a Cournot duopoly in which the firms face a different market demand in each of two scenarios, and make their output decisions before uncertainty is resolved. The way in which firms value the possible outcomes is critical when deciding their strategies. In real-life situations the attitudes that agents exhibit can vary from extreme pessimism to extreme optimism, and it is possible to characterize their behavior according to their degrees of optimism. In this context, we identify the sets of equilibria for the full range of degrees of optimism, and illustrate the results with the analysis of some cases in which the demand functions are linear.