This paper introduces the simple and standard Cournot duopoly model facing the asymmetric situation in which firm 1 has partial information and firm 2 has perfect information on the cost function of their rival under different risk-taking behaviors. We present two types of risk-taking behaviors of firms to analyze this asymmetric situation. First, when both firms are risk-neutral, compared with the results that each firm has perfect information on its rival’s cost function, the optimal output level of firm 2 does not change; however, that of firm 1 can increase or decrease. Second, we consider that firm 1 is risk-averse and firm 2 is risk-neutral. We find that firm 1’s optimal output level decreases and that of firm 2 increases under this kind of asymmetric information, compared with the results in the case in which both firms are risk-neutral. Further, we find that firms’ optimal strategies depend on their risk-taking behaviors.