The rapid growth of live-streaming has transformed traditional e-commerce into an interactive and immersive experience, giving birth to live-streaming e-commerce. This paper investigates the strategic interactions between brands, social media influencers, and consumers under this mechanism. Using evolutionary game theory, we model decision-making dynamics across these three parties and analyze how their strategies develop over time. Our framework incorporates contractual penalties between brands and influencers, rewards for influencers, product returns, and subscription fees to capture realistic market behaviors. We derive replicator dynamics equations for each participant group and identify stable equilibrium strategies for the entire system. The application of replicator dynamics offers valuable perspectives on temporary states and strategies that achieve long-term equilibrium. We also present numerical simulations to validate the effectiveness of our model. In addition, we show how parameters, such as penalties and rewards, influence strategy selection and allow the system to achieve stability successfully. This research provides actionable recommendations for optimizing partnerships in live-streaming e-commerce supply chains.