This research investigates the optimal punitive damages relating the intellectual property under different circumstances via the Stackelberg game approach. Unlike traditional compensatory damages, punitive damages can compensate the victim more than the actual damages caused by the tortfeasor. In this research, we examine the mechanism of punitive damages for online copyright infringement and decide the optimal punitive damages based on the illegal benefits of the copyright tortfeasor. Our research aims to inspect the potential effect of the determined punitive damages with respect to the optimal social welfare via the framework of the Stackelberg game, in which the government is the leader and other firms, including studio, legal platform, and illegal platform, are the followers. We solve the Stackelberg game model by the backward induction method and gradient ascent with momentum for those four models purposed in our study. We check several scenarios through our proposing models, we find that the increase in the quality of the piracy also increases the optimal level of the punitive damages, and this fact aligns with what the public believe. The former two models recommend the government impose higher penalty when the quality of the unit advertising profit of the illegal platform is low but the latter two model contrarily recommend the government impose higher penalty when the quality of the unit advertising profit of the illegal platform is good to increase the social welfare.