With the extensive use of machine learning technologies, data providers encounter increasing privacy risks. Recent legislation, such as GDPR, obligates organizations to remove requested data and its influence from a trained model. Machine unlearning is an emerging technique designed to enable machine learning models to erase users' private information. Although several efficient machine unlearning schemes have been proposed, these methods still have limitations. First, removing the contributions of partial data may lead to model performance degradation. Second, discrepancies between the original and generated unlearned models can be exploited by attackers to obtain target sample's information, resulting in additional privacy leakage risks. To address above challenges, we proposed a game-theoretic machine unlearning algorithm that simulates the competitive relationship between unlearning performance and privacy protection. This algorithm comprises unlearning and privacy modules. The unlearning module possesses a loss function composed of model distance and classification error, which is used to derive the optimal strategy. The privacy module aims to make it difficult for an attacker to infer membership information from the unlearned data, thereby reducing the privacy leakage risk during the unlearning process. Additionally, the experimental results on real-world datasets demonstrate that this game-theoretic unlearning algorithm's effectiveness and its ability to generate an unlearned model with a performance similar to that of the retrained one while mitigating extra privacy leakage risks.