# Differential Privacy in Federated Learning: An Evolutionary Game Analysis
## 2 Related Work and Preliminaries
### Incentive Mechanisms in Federated Learning  
In federated learning (FL), clients are rational actors who face critical challenges, such as potential privacy leakage risks, when participating in collaborative model training. These inherent costs create disincentives for client participation, leading to situations where a subset of clients may be reluctant to engage in the collaborative learning process. To address this fundamental challenge, the implementation of incentive mechanisms becomes essential. Specifically, a rationally structured incentive scheme can reward clients commensurate with their contributions, thereby motivating them to tolerate associated costs and actively participate in FL frameworks. Such mechanisms aim to align individual rationality with collective learning objectives by balancing computational costs, privacy risks, and model performance. Significant research efforts have been devoted to developing optimized solutions from diverse perspectives. For instance, [PERSON] et al. [30] proposed a reputation-rating algorithm to evaluate client trustworthiness, thereby enhancing global model performance through rigorous credibility assessment. [PERSON] et al. [31] introduced a fair and robust federated learning (FRFL) framework, leveraging contract theory principles to design a client selection mechanism that mitigates free-riding behaviors while ensuring reliability. [PERSON] et al. [32] developed DeepChain, a blockchain-enabled deep learning architecture that utilizes value-driven incentives to align client behaviors with collaborative objectives through cryptographic commitments. However, these seminal publications predominantly overlook the critical aspect of inter-client strategic interactions. As federated learning fundamentally constitutes an iterative multi-round process, it inherently embodies evolutionary game-theoretic characteristics, where clients dynamically adapt their training strategies in response to environmental feedback (e.g., reward distributions and peer behaviors). Evolutionary game theory provides essential analytical tools, such as replicator dynamics equations and evolutionarily stable strategies, which enable the establishment of a realistic behavioral modeling framework tailored to FL ecosystems. This paradigm captures continuous strategy evolution through population-level adaptation mechanisms, transcending the limitations of conventional static game models. For example, [PERSON] et al. [33] constructed an evolutionary game model to analyze the long-term strategies of device participation in training, considering the resource heterogeneity of mobile devices. Their study revealed the driving mechanism of decision evolution in balancing resource constraints and benefits. In a decentralized setting, [PERSON] et al. [34] proposed a blockchain-based federated learning (BCFL) game framework. Through a two-stage \"node selection-data sharing\" strategy game, they demonstrated how blockchain transparency facilitates trust-building among participants and derived the conditions for cooperative benefit allocation under Nash equilibrium. Regarding the stability of complex federated networks, [PERSON] et al. [35] designed a cloud federation formation model that integrates genetic algorithms with evolutionary games. Their findings proved that multi-round strategy adjustments can converge to a federated topology optimized for quality of service (QoS). In terms of security defense, [PERSON] et al. [36] incorporated the probability of IIoT network attacks into the game payoff function and proposed the FedGame model. This model leverages evolutionary stable strategies (ESSs) to filter out collaborative alliances capable of resisting malicious node interference. These studies offer a systematic framework for designing incentive mechanisms in federated learning across multiple dimensions, including resource allocation, trust-building, topology optimization, and security defense.