# Differential Privacy in Federated Learning: An Evolutionary Game Analysis
## 1 Introduction  
In the era of the information age, the generation and accumulation of data is accelerating at an unprecedented rate. However, this vast amount of data is often fragmented across various locations and managed by multiple entities. Traditionally, analyzing such dispersed data has relied heavily on centralized processing, which requires the aggregation of data from various locations. As concerns over data security and privacy continue to grow, conventional approaches have become increasingly inadequate, often resulting in significant privacy leakages [1]. Specifically, when private data are used for model training, the trained models may inadvertently disclose sensitive information through their behavior or structure [2, 3, 4]. In recent years, federated learning (FL) has emerged as a novel machine learning (ML) paradigm, gradually gaining traction. The core idea of federated learning is to enable multiple clients to collaboratively train a shared machine learning model while keeping their data decentralized and locally stored. Although federated learning can protect client privacy to some extent, significant challenges remain. For example, [PERSON] et al. [5] introduced membership inference attacks, highlighting a significant privacy vulnerability within federated learning. In response to such challenges, recent studies have increasingly turned to advanced cryptographic tools, such as secure multi-party computation and differential privacy, to enhance privacy protection in federated learning [6]. For instance, [PERSON] et al. [7] investigated the trade-off between convergence performance and differential privacy protection in federated learning under a unified noise scale. [PERSON] et al. [8] examined the relationship between model accuracy and privacy protection with a uniform noise scale. [PERSON] et al. [9] studied the relationship between privacy loss and model performance. However, despite significant progress in addressing privacy concerns in federated learning, many critical questions remain open for exploration. For instance, while existing studies have highlighted the importance of privacy protection through techniques such as secure multi-party computation and differential privacy, there is still a need for more comprehensive frameworks to quantify privacy leakage and to understand the impact of privacy preferences on model performance. This gap underscores the ongoing and evolving nature of privacy protection in federated learning, making it a highly relevant and active area of research. Game theory, particularly evolutionary game theory, provides a powerful framework for analyzing the strategic behavior of participants in federated learning systems. Traditional game theory focuses on systematically analyzing competitors to choose the best strategy for maximizing personal benefits [10]. In contrast, evolutionary game theory [11; 12; 13] emphasizes the evolution of strategies and changes at the population level. It is widely used for analyzing interactions between groups. In the context of federated learning, evolutionary game theory can effectively model the dynamic behavior of clients as they balance the trade-off between privacy protection and model performance. Clients aim to maximize their utility by selecting strategies that protect their privacy while still contributing to the global model's accuracy. The concept of population share in evolutionary game theory can be used to reflect the proportion of clients adopting different strategies over time, capturing the evolving dynamics of client behavior [14]. Recently, incentive allocation mechanisms in evolutionary game theory have garnered academic attention [15; 16; 17]. A well-designed incentive mechanism can shape the behavioral patterns of the group during evolutionary games, reduce the selfish behavior of individuals, and thereby enhance group stability. Previous studies [18; 19; 20] have applied game theory ideas using various models but have not considered the impact of individual strategies on overall incentives.