<p>In the digital era, the widespread dissemination of rumors on online social networks (OSNs) has emerged as a formidable challenge. Driven by the pursuit of user engagement, competitive social platforms often overlook effective rumor control mechanisms, exacerbating information disorders. A framework for modeling platform-wide competition during short-term but explosive information transmission scenarios is proposed in this paper. We first introduce a novel dynamic susceptible, exposed, infected, labeled, and recovered (SEILR) model, which is meticulously designed to capture the intricate dynamics of information transmission across multilayer networks. Second, we formulate a zero-sum differential game framework to analyze the strategic interactions among competing platforms. By integrating critical cost factors, including user losses, potential government sanctions, and social management expenses, our framework provides a comprehensive understanding of the economic implications of rumor control. Third, to address the complexity of deriving optimal strategies within the proposed framework, we innovatively combine iterative approximate dynamic programming (ADP) with neural networks. Leveraging the approximation capabilities of the neural networks, we enhance the efficiency and accuracy of the ADP process. Through extensive numerical simulations conducted in a duopoly market setting, we validate the efficacy of our proposed strategies. The results demonstrate their remarkable cost efficiency and superior rumor management capabilities relative to those of the existing approaches.</p>