In order to increase market share and enhance business competitiveness, comprehensive retail platforms (CRPs), such as Taobao.com and JD.com, are currently cooperating with social service platforms (SSPs), such as TikTok and Meiyou. Therefore, more research attention needs to be paid to such platform supply chains. To this end, this study develops stylized models to explore the impacts of the market power of an SSP and the precision marketing of a CRP on platform cooperation. Our major research findings include the following. First, we investigate the equilibrium strategy and find that if the CRPs commission rate is low (high), cooperation becomes the equilibrium strategy when the SSPs market power is low (high). Second, in most cases, market power will affect the equilibrium strategy with a change in the commission rate. However, in a certain range of commission rates, the equilibrium strategy remains unchanged. That is, market power has an "invalidation effect" on the equilibrium strategy. Finally, we find that as the commission rate increases, precision marketing reduces the willingness of supply chain members to implement platform cooperation. However, when the commission rate is low, precision marketing stimulates cooperation between platforms. The study results provide guidelines for improving the cooperation between CRPs and SSPs.