# Privacy-Preserving Monotonicity of Differential Privacy Mechanisms
## 6 Privacy-Preserving Monotonicity of Several Differential Privacy Mechanisms
### Privacy-Preserving Monotonicity Based on Expected Estimation Error  
**Theorem 1**. : _The \(EEE\) of Laplace mechanism decreases as privacy budget \(\varepsilon\) increases._  
Proof. : The probability density function of Laplace distribution is  
\[p(x)=\frac{1}{2b}\exp(-\frac{|x|}{b}) \tag{17}\]  
where \(b=\frac{\Delta f}{\varepsilon}\). Then  
\[EEE=\int_{-\infty}^{+\infty}\frac{1}{2b}\exp(-\frac{|x|}{b})|x|dx=-\int_{- \infty}^{0}\frac{1}{2b}\exp(\frac{x}{b})xdx+\int_{0}^{+\infty}\frac{1}{2b} \exp(\frac{-x}{b})xdx=b=\frac{\Delta f}{\varepsilon} \tag{18}\]  
Therefore, \(\frac{dEEE}{dt}=-\frac{\Delta f}{\varepsilon^{2}}<0\). The \(EEE\) of Laplace mechanism decreases as privacy budget \(\varepsilon\) increases. The characteristic functions of Laplace mechanism can refer to literature [64] for analysis of utility metric with privacy budget varying. **Corollary 1**. : _The modulus \(|\phi(t)|\) of characteristic function of Laplace mechanism increases as privacy budget \(\varepsilon\) increases._  
Proof. : Since the characteristic function of Laplace mechanism is  
\[\phi(t)=\frac{\varepsilon^{2}}{\varepsilon^{2}+\Delta f^{2}t^{2}} \tag{19}\]  
the modulus of characteristic function of Laplace mechanism is  
\[|\phi(t)|=\frac{\varepsilon^{2}}{\varepsilon^{2}+\Delta f^{2}t^{2}} \tag{20}\]Thus, we can get  
\[\frac{d(|\phi(t)|)}{d\varepsilon}=\frac{2\varepsilon\Delta f^{2}t^{2}}{( \varepsilon^{2}+\Delta f^{2}t^{2})^{2}}\geq 0 \tag{21}\]  
Therefore, the modulus \(|\phi(t)|\) of characteristic function of Laplace mechanism increases as privacy budget \(\varepsilon\) increases. By Theorem 1, the expected estimation error of Laplace mechanism decreases as privacy budget increases. The privacy-preserving level of Laplace mechanism decreases as privacy budget increases. Corollary 1 shows that the modulus of characteristic function of Laplace mechanism increases as privacy budget increases. Thus, the data utility using Laplace mechanism increases as privacy budget increases. According to Definition 8, Theorem 1 shows the privacy-preserving monotonicity of Laplace mechanism. Privacy-preserving monotonicity of Laplace mechanism shows the trade-off between privacy and utility according to the Theorem 1 and Corollary 1. The \(EEE\) of Gaussian mechanism decreases as privacy budget \(\varepsilon\) increases. Proof. : The probability density function of Gaussian distribution is  
\[p(x)=\frac{1}{\sqrt{2\pi\sigma}}\exp(-\frac{x^{2}}{2\sigma^{2}}) \tag{22}\]  
where \(\sigma=\frac{\Delta_{2}f\sqrt{2\ln(\frac{125}{\delta})}}{\varepsilon}\). Then  
\[EEE=\int_{-\infty}^{+\infty}\frac{1}{\sqrt{2\pi\sigma}}\exp(-\frac{x^{2}}{2 \sigma^{2}})|x|dx=\frac{\sqrt{2}\sigma}{\sqrt{\pi}}=\frac{2\Delta_{2}f\sqrt{ \ln(\frac{125}{\delta})}}{\varepsilon\sqrt{\pi}} \tag{23}\]  
Therefore, \(\frac{dEEE}{d\varepsilon}=-\frac{2\Delta_{2}f\sqrt{\ln(\frac{125}{\delta})}}{ \varepsilon^{2}\sqrt{\pi}}<0\). The \(EEE\) of Gaussian mechanism decreases as privacy budget \(\varepsilon\) increases. The characteristic functions of Gaussian mechanism can refer to literature [64] for analysis of utility metric with privacy budget varying. The modulus \(|\phi(t)|\) of characteristic function of Gaussian mechanism decreases as privacy budget \(\varepsilon\) increases. Proof. : Since the characteristic function of Gaussian mechanism is  
\[\phi(t)=\exp(-\frac{\Delta_{2}f^{2}t^{2}\ln(\frac{125}{\delta})}{\varepsilon^{ 2}}) \tag{24}\]  
the modulus of characteristic function of Gaussian mechanism is  
\[|\phi(t)|=\exp(-\frac{\Delta_{2}f^{2}t^{2}\ln(\frac{125}{\delta})}{\varepsilon ^{2}}) \tag{25}\]  
Thus, we can get  
\[\frac{d(|\phi(t)|)}{d\varepsilon}=\frac{2\Delta_{2}f^{2}t^{2}\ln(\frac{125}{ \delta})}{\varepsilon^{3}}\exp(-\frac{\Delta_{2}f^{2}t^{2}\ln(\frac{125}{ \delta})}{\varepsilon^{2}})\geq 0 \tag{26}\]  
Therefore, the modulus \(|\phi(t)|\) of characteristic function of Gaussian mechanism increases as privacy budget \(\varepsilon\) increases.