# Privacy-Preserving Monotonicity of Differential Privacy Mechanisms
## 6 Privacy-Preserving Monotonicity of Several Differential Privacy Mechanisms
### Privacy-Preserving Monotonicity Based on Entropy  
To prove privacy-preserving monotonicity of exponential mechanism, we introduce the following Cauchy-Schwarz inequality [65]. **Lemma 1** (Cauchy-Schwarz Inequality). : _Let \((a_{1},a_{2},\ldots,a_{n})\) and \((b_{1},b_{2},\ldots,b_{n})\) be two sequences of real numbers. Then_  
\[(\sum_{i=1}^{n}a_{i}b_{i})^{2}\geq\sum_{i=1}^{n}a_{i}  
we have  
\[ \begin{split}\frac{dH(\varepsilon)}{d\varepsilon}&=- \frac{1}{(\sum_{j=1}^{n}e^{\varepsilon u_{j}})^{2}\ln 2}\sum_{i=1}^{n}e^{ \varepsilon u_{i}}\sum_{j=1}^{n}\varepsilon u_{i}(u_{i}-u_{j})e^{\varepsilon u _{j}}\\ &=-\frac{\varepsilon}{(\sum_{j=1}^{n}e^{\varepsilon u_{j}})^{2} \ln 2}(\sum_{i=1}^{n}u_{i}^{2}e^{\varepsilon u_{i}}\sum_{i=1}^{n}e^{\varepsilon u _{i}}-(\sum_{i=1}^{n}u_{i}e^{\varepsilon u_{i}})^{2})\end{split}  \tag{33}\]  
According to Lemma 1, we get  
\[\sum_{i=1}^{n}u_{i}^{2}e^{\varepsilon u_{i}}\sum_{i=1}^{n}e^{ \varepsilon u_{i}}=\sum_{i=1}^{n}(\sqrt{u_{i}^{2}e^{\varepsilon u_{i}}})^{2} \sum_{i=1}^{n}(\sqrt{e^{\varepsilon u_{i}}})^{2}\geq(\sum_{i=1}^{n}(\sqrt{u_{ i}^{2}e^{\varepsilon u_{i}}}\sqrt{e^{\varepsilon u_{i}}}))^{2}=(\sum_{i=1}^{n}u_{i}e^{ \varepsilon u_{i}})^{2} \tag{34}\]  
Therefore, \(\frac{dH(\varepsilon)}{d\varepsilon}\leq 0\). The \(ENT\) of output of exponential mechanism decreases as privacy budget \(\varepsilon\) increases. **Corollary 3**. : _The utility metric \(1-NE\) of output of exponential mechanism increases as privacy budget \(\varepsilon\) increases._  
Proof. : According to Equation (29), the normalized entropy of output of exponential mechanism is  
\[NE=\frac{H(\varepsilon)}{\log_{2}n}=-\frac{1}{\log_{2}n}\sum_{i=1}^{n}\frac{e^ {\varepsilon u_{i}}}{\sum_{j=1}^{n}e^{\varepsilon u_{j}}}\log_{2}\frac{e^{ \varepsilon u_{i}}}{\sum_{j=1}^{n}e^{\varepsilon u_{j}}} \tag{35}\]  
Since \(\frac{dH(\varepsilon)}{d\varepsilon}\leq 0\), the normalized entropy \(NE\) of output of exponential mechanism decreases as privacy budget \(\varepsilon\) increases. Therefore, the utility metric \(1-NE\) of output of exponential mechanism increases as privacy budget \(\varepsilon\) increases. By Theorem 3, the entropy of output of exponential mechanism decreases as privacy budget increases. The privacy-preserving level of output of exponential mechanism decreases as privacy budget increases. According to the Corollary 3, the utility metric \(1-NE\) of output of exponential mechanism increases as privacy budget increases. Thus, the data utility using exponential mechanism increases as privacy budget increases. According to Definition 8, Theorem 3 shows the privacy-preserving monotonicity of exponential mechanism. Privacy-preserving monotonicity shows the trade-off between privacy and utility based on Theorem 3 and Corollary 3. **Theorem 4**. : _The \(ENT\) of output of bivariate randomized response mechanism decreases as privacy budget \(\varepsilon\) increases._  
Proof. : For any possible output \(\xi\in\{0,1\}\) and any input \(b\in\{0,1\}\), the conditional probability matrix of the bivariate randomized response mechanism is Equation (12). Let \(\pi_{1}\) denotes the proportion of value \(1\) in \(\{0,1\}\). Then, the proportion of value \(0\) is \(\pi_{0}=1-\pi_{1}\).