# Differential Privacy in Federated Learning: An Evolutionary Game Analysis
## 4 Game Analysis  
In this section, we investigate the existence and stability of the replicator dynamics described by Equation (9). Regarding existence, we demonstrate that the game has at least one stable point (equilibrium). Regarding stability, we show that once the game reaches equilibrium, it can return to the equilibrium state even if disturbed. First, we prove the boundedness of Equation (9) in Lemma 1. **Lemma 1**: _For all \(n,v\in\{1,\ldots,N\}\), the first-order derivative of \(\phi_{n}(x_{1},\ldots,x_{N})\) with respect to \(x_{v}(t)\) is bounded._  
Proof of Lemma 1.: For ease of calculation, we omit \(t\) in the proof. Then, the coefficients of the first, second, and third terms in Equation (3) are constants. The sum of the first and second terms is denoted by \(B\), and the coefficient of the third term is denoted by \(C\). Thus, Equation (3) can be rewritten as  
\[L_{g}=B+C\cdot\sum_{i=1}^{N}(\sigma_{n}^{2}\cdot I\cdot x_{i}) \tag{10}\]The derivative of \(\phi_{n}(x_{1},\ldots,x_{N})\) with respect to \(x_{v}\) can be given as follows:  
\[\frac{d\phi_{n}(x_{1},\ldots,x_{N})}{dx_{v}}=\Delta\bigg{[}\frac{dx_{n}}{x_{v}}( U_{n}-\bar{U})+x_{n}\bigg{(}\frac{dU_{n}}{dx_{v}}-\frac{d\bar{U}}{dx_{v}}\bigg{)} \bigg{]}. \tag{11}\]  
This equation represents the partial derivative of the replicator dynamics equation \(\phi_{n}(x_{1},\ldots,x_{N})\) with respect to the population share of strategy \(v\), denoted as \(x_{v}\). It is derived using the product rule of differentiation. Specifically, considering \(\phi_{n}\) as a product of two terms, \(\Delta x_{n}\) and \((U_{n}-\bar{U})\), its derivative with respect to \(x_{v}\) is given by \(\frac{d\phi_{n}}{dx_{v}}=\frac{d(\Delta x_{n})}{dx_{v}}(U_{n}-\bar{U})+( \Delta x_{n})\frac{d(U_{n}-\bar{U})}{dx_{v}}\). To simplify the notation, we define  
\[G(x_{1},\ldots,x_{N})=\sum_{i=1}^{N}(\sigma_{n}^{2}\cdot I\cdot x_{i}),\quad H (x_{1},\ldots,x_{N})=I\cdot\sum_{i=1}^{N}\frac{x_{i}}{\sigma_{i}}. \tag{12}\]  
Thus, we can obtain  
\[\frac{dU_{n}}{dx_{v}}=\frac{-r\cdot\frac{dG(x_{1},\ldots,x_{N})}{dx_{v}}}{ \big{[}B+c\cdot G(x_{1},\ldots,x_{N})\big{]}^{2}}\cdot\frac{1}{\sigma_{n}H(x_{ 1},\ldots,x_{N})}+\frac{r}{B+c\cdot G(x_{1},\ldots,x_{N})}\cdot\frac{\frac{dH( x_{1},\ldots,x_{N})}{dx_{v}}}{\sigma_{n}H(x_{1},\ldots,x_{N})^{2}}. \tag{13}\]  
Since \(\frac{dG(x_{1},\ldots,x_{N})}{dx_{v}}=\sigma_{v}^{2}\cdot I\), \(\frac{dH(x_{1},\ldots,x_{N})}{dx_{v}}=\frac{I}{\sigma_{v}}\), we can obtain that \(\Big{|}\frac{dU_{n}}{dx_{v}}\Big{|}\) is bounded. Hence, \(\Big{|}\frac{d\phi_{n}(x_{1},\ldots,x_{N})}{dx_{v}}\Big{|}\) is also bounded. Then, we can prove that the solution of (9) exists. **Theorem 1**. : _For all \(n\in\{1,\ldots,N\}\), the solution of (9) exists._  
Proof of Theorem 1.: From Lemma 1, we can obtain that \(\Big{|}\frac{d\phi_{n}(x_{1},\ldots,x_{N})}{dx_{v}}\Big{|}\) is bounded. In addition, it is straightforward to see that \(\phi_{n}(x_{1},\ldots,x_{N})\) and \(\frac{d\phi_{n}(x_{1},\ldots,x_{N})}{dx_{v}}\) are both continuous. Therefore, \(\phi_{n}(x_{1},\ldots,x_{N})\) satisfies the Lipschitz condition with respect to \(x_{1},\ldots,x_{N}\). According to the Picard-Lindelof theorem, for any given initial conditions, the system of differential equations has a unique solution in the region under consideration. Now, we analyze the stability of this evolutionary game. We begin with the definition of Nash equilibrium. **Definition 1**.