# Differential Privacy in Federated Learning: An Evolutionary Game Analysis
## 3 System Model and Evolutionary Game Formulation  
### System Model  
We consider a differential privacy federated learning model comprising \(I\) clients, denoted by \(P_{1},\ldots,P_{I}\), and a central server. When the server initiates a training task, it assigns the task to the clients and offers incentives to those who participate. Clients, in turn, decide whether to engage in the training task and, if so, employ differential privacy mechanisms to protect the privacy of their local data. Specifically, they select appropriate numbers of local iterations, \(E_{i}\), and add Gaussian noise with power \(\sigma_{i}\) during model training and parameter uploading. The federated learning system model is illustrated in Figure 3.  
To analyze this model within the framework of evolutionary game theory, we define the following components:  
1. Players: The players in this game are the clients, denoted by \(P_{1},\ldots,P_{I}\). 2. Strategies: The \(n\)th strategy available to a player can be expressed as \(S_{n}=(E_{n},\sigma_{n})\), where \(E_{n}\) is the number of local iterations per round, and \(\sigma_{n}^{2}\) is the Gaussian noise power. We assume there are \(N\) strategies in total, denoted by \(S_{1},\ldots,S_{N}\). Here,'strategies' represent the different combinations of local iterations and Gaussian noise power that a client can choose. These strategies are formulated based on the practical application of differential privacy in FL. Different noise powers represent varying levels of privacy protection, with higher noise offering stronger privacy but potentially impacting model accuracy and, thus, the incentive pool. The number of local iterations also influences both model contribution and potential privacy leakage. These strategies are relevant to clients because they directly control the trade-off between their privacy protection and their potential utility. Clients with different privacy preferences or data sensitivity levels might choose different strategies. Figure 3: Federated learning system model. Population share: The population share represents the proportion of clients choosing a particular strategy within the strategy space. Let \(x_{n}\in\mathbf{X}\) denote the proportion of the population that selects strategy \(S_{n}\), and it satisfies \(\sum_{n=1}^{N}x_{n}=1\). 4. Utility: The utility for each client is determined by their net income, which is the difference between the payoff obtained from the server and the training costs incurred. The utility for selecting strategy \(S_{n}\) at time \(t\) is denoted by \(U_{n}(t)\). For simplicity, this is subsequently abbreviated as \(U_{n}\). In the framework of evolutionary game theory, each player can maximize their utility by adjusting their own strategy. The selection of strategies by clients is of significant importance because it directly impacts the trade-off between utility and privacy. An optimal strategy aims to maximize client utility while adhering to privacy constraints. For instance, a client prioritizing incentives above all else might choose a strategy that introduces minimal or no noise to their data. Although this could potentially maximize their contribution to an incentive pool, it would simultaneously lead to a substantial increase in privacy leakage, potentially undermining the overall privacy guarantees of the system. Conversely, if all clients were to adopt overly conservative strategies by adding excessive noise, the aggregated model would suffer from poor training quality, resulting in a diminished incentive pool and, ultimately, reduced utility for all clients. Therefore, the strategic selection of privacy mechanisms is crucial to achieving a delicate equilibrium, enabling clients to derive meaningful utility from participation while simultaneously ensuring robust privacy protection. In essence, effective strategy selection is key to balancing client incentives with the fundamental requirement of privacy preservation within the system.