This paper examines federated learning, a decentralized machine learning paradigm, focusing on privacy challenges. We introduce differential privacy mechanisms to protect privacy and quantify their impact on global model performance. Using evolutionary game theory, we establish a framework to analyze strategy dynamics and define utilities for different strategies based on Gaussian noise powers and training iterations. A differential privacy federated learning model (DPFLM) is analyzed within this framework. A key contribution is the thorough existence and stability analysis, identifying evolutionarily stable strategies (ESSs) and confirming their stability through simulations. This research provides theoretical insights for enhancing privacy protection in federated learning systems.