# Privacy-Preserving Monotonicity of Differential Privacy Mechanisms
## 6 Privacy-Preserving Monotonicity of Several Differential Privacy Mechanisms
### Privacy-Preserving Monotonicity Based on Entropy
** According to Equation (45), the normalized entropy of output of multivariate randomized response mechanism is  
\[NE=\frac{H(\varepsilon)}{\log_{2}n}=-\frac{1}{\log_{2}n}\sum_{i=1}^{n}\frac{e^ {\varepsilon u_{i}}}{\sum_{j=1}^{n}e^{\varepsilon u_{j}}}\log_{2}\frac{e^{ \varepsilon u_{i}}}{\sum_{j=1}^{n}e^{\varepsilon u_{j}}} \tag{35}\]0  
Since \(\frac{dH_{\xi}(\epsilon)}{d\epsilon}\leq 0\), the normalized entropy \(NE\) of output of multivariate randomized response mechanism decreases as privacy budget \(\epsilon\) increases. Therefore, the utility metric \(1-NE\) of output of multivariate randomized response mechanism increases as privacy budget \(\epsilon\) increases. By Theorem 5, the entropy of output of multivariate randomized response mechanism decreases as privacy budget increases. The privacy-preserving level of output of multivariate randomized response mechanism decreases as privacy budget increases. According to the Corollary 5, the utility metric of output of multivariate randomized response mechanism increases as privacy budget increases. Thus, the data utility using multivariate randomized response mechanism increases as privacy budget increases. According to Definition 8, Theorem 5 shows the privacy-preserving monotonicity of multivariate randomized response mechanism. Privacy-preserving monotonicity of multivariate randomized response mechanism demonstrates the trade-off between privacy and utility based on Theorem 5 and Corollary 5. Furthermore, Laplace mechanism and Gaussian mechanism are applicable for categorical data based on the entropy of continuous random variables. Since the entropy of output of Laplace mechanism and Gaussian mechanism may be negative, we do not consider the variant \(1-NE\) of normalized entropy as the utility metric of Laplace mechanism and Gaussian mechanism. We only use the modulus of characteristic function as the utility metric of Laplace mechanism and Gaussian mechanism. Thus, we can get the following theorems. **Theorem 6**. : _The \(ENT\) of output of Laplace mechanism decreases as privacy budget \(\epsilon\) increases._  
**Proof. ** The entropy of output of Laplace mechanism is  
\[NE=\frac{H(\varepsilon)}{\log_{2}n}=-\frac{1}{\log_{2}n}\sum_{i=1}^{n}\frac{e^ {\varepsilon u_{i}}}{\sum_{j=1}^{n}e^{\varepsilon u_{j}}}\log_{2}\frac{e^{ \varepsilon u_{i}}}{\sum_{j=1}^{n}e^{\varepsilon u_{j}}} \tag{35}\]1  
Therefore, \(\frac{dENT}{d\epsilon}=-\frac{1}{\epsilon\ln 2}<0\). The \(ENT\) of output of Laplace mechanism decreases as privacy budget \(\epsilon\) increases. According to the Theorem 6, the entropy of output of Laplace mechanism decreases as privacy budget increases. The privacy-preserving level of output of Laplace mechanism decreases as privacy budget increases. According to Definition 8, Theorem 6 demonstrates the privacy-preserving monotonicity of Laplace mechanism. Similarly, privacy-preserving monotonicity of Laplace mechanism demonstrates the trade-off between privacy and utility based on Theorem 6 and Corollary 1. **Theorem 7**. : _The \(ENT\) of output of Gaussian mechanism decreases as privacy budget \(\epsilon\) increases._  
**Proof. ** The entropy of output of Gaussian mechanism is  
\[NE=\frac{H(\varepsilon)}{\log_{2}n}=-\frac{1}{\log_{2}n}\sum_{i=1}^{n}\frac{e^ {\varepsilon u_{i}}}{\sum_{j=1}^{n}e^{\varepsilon u_{j}}}\log_{2}\frac{e^{ \varepsilon u_{i}}}{\sum_{j=1}^{n}e^{\varepsilon u_{j}}} \tag{35}\]2  
Therefore, \(\frac{dENT}{d\epsilon}=-\frac{1}{\epsilon\ln 2}<0\). The \(ENT\) of output of Gaussian mechanism decreases as privacy budget \(\epsilon\) increases. By Theorem 7, the entropy of output of Gaussian mechanism decreases as privacy budget increases. The privacy-preserving level of output of Gaussian mechanism decreases as privacy budget increases. According to Definition 8, Theorem 7 demonstrates the privacy-preserving monotonicity of Gaussian mechanism. Similarly, privacy-preserving monotonicity of Gaussian mechanism demonstrates the trade-off between privacy and utility based on Theorem 7 and Corollary 2.