We study the R&D performance of Cournot aligopolists. To this end we model a one-shot noncooperative game in which firms invest in R&D, with the aim of being first in an uncertain competition for a patentable cost-reducing innovation. The incentives to innovate are market profits and not exogenously given prizes as in most of the earlier literature. Thus, incentives depend on the number of firms. We show that increasing rivalry may increase or decrease the individual R&D espenditure, there may be underinvestment with the various effect to the socially optimal level. Moreover, we identify the various effects which are responsible for the difference between our results and some conclusions of closely related contributions.