A rumor about an entity can spread rapidly through online social networks (OSNs), which may lead to serious consequences. This article focuses on developing a cost-effective rumor-refuting strategy in the situation where the rumormonger is strategic. Based on a novel individual-based rumor spreading model, we estimate the expected net benefit of the rumormonger as well as the expected total loss of the rumor victim. On this basis, we reduce the original problem to a differential game-theoretic model. Next, we derive a system for solving the model. By solving the system, we get a dynamic strategy pair. Through extensive comparative experiments, we find that this strategy pair is effective in terms of the solution concept of Nash equilibrium. Hence, we conclude that the rumor-refuting strategy contained in this strategy pair is cost-effective. Finally, we examine the effect of the structure of the OSN on the cost effectiveness of the rumor-supporting and rumor-refuting strategies contained in the proposed strategy pair. To our knowledge, this is the first time the rumor-refuting problem is addressed under the assumption that the rumormonger is strategic.