# Privacy-Preserving Monotonicity of Differential Privacy Mechanisms
## 6 Privacy-Preserving Monotonicity of Several Differential Privacy Mechanisms
### Privacy-Preserving Monotonicity Based on Entropy
Then, we have  
\[P(\xi=0)=(1-\pi_{1})\frac{e^{\varepsilon}}{1+e^{\varepsilon}}+\pi_{1}\frac{1}{ 1+e^{\varepsilon}}=\frac{(1-\pi_{1})e^{\varepsilon}+\pi_{1}}{1+e^{\varepsilon}} \tag{36}\]  
and  
\[P(\xi=1)=(1-\pi_{1})\frac{1}{1+e^{\varepsilon}}+\pi_{1}\frac{e^{\varepsilon}}{ 1+e^{\varepsilon}}=\frac{1-\pi_{1}+\pi_{1}e^{\varepsilon}}{1+e^{\varepsilon}} \tag{37}\]Thus, we get  
\[H_{\xi}(\varepsilon) =-P(\xi=0)\log_{2}P(\xi=0)-P(\xi=1)\log_{2}P(\xi=1)\] \[=-\frac{(1-\pi_{1})e^{\varepsilon}+\pi_{1}}{1+e^{\varepsilon}}\log _{2}\frac{(1-\pi_{1})e^{\varepsilon}+\pi_{1}}{1+e^{\varepsilon}}-\frac{(1-\pi_ {1})+\pi_{1}e^{\varepsilon}}{1+e^{\varepsilon}}\log_{2}\frac{(1-\pi_{1})+\pi_{ 1}e^{\varepsilon}}{1+e^{\varepsilon}} \tag{38}\]  
and  
\[\frac{dH_{\xi}(\varepsilon)}{d\varepsilon}=\frac{(2\pi_{1}-1)e^{\varepsilon} }{(1+e^{\varepsilon})^{2}}\log_{2}\frac{e^{\varepsilon}(1-\pi_{1})+\pi_{1}}{ \pi_{1}e^{\varepsilon}+1-\pi_{1}} \tag{39}\]  
Since \(\pi_{1}e^{\varepsilon}+1-\pi_{1}>0\), we have  
\[e^{\varepsilon}(1-\pi_{1})+\pi_{1}>\pi_{1}e^{\varepsilon}+1-\pi_{1}\Leftrightarrow 2 \pi_{1}(1-e^{\varepsilon})>1-e^{\varepsilon} \tag{40}\]  
Obviously, \(1-e^{\varepsilon}<0\). Thus, we have  
\[2\pi_{1}(1-e^{\varepsilon})>1-e^{\varepsilon}\Leftrightarrow\pi_{1}<\frac{1}{2} \tag{41}\]  
Similarly, \(0<\frac{e^{\varepsilon}(1-\pi_{1})+\pi_{1}}{\pi_{1}e^{\varepsilon}+1-\pi_{1}}< 1\Leftrightarrow\frac{1}{2}\leq\pi_{1}\leq 1\). Therefore, when \(0\leq\pi_{1}\leq\frac{1}{2}\), we have \(2\pi_{1}-1\leq 0\) and \(\log_{2}\frac{e^{\varepsilon}(1-\pi_{1})+\pi_{1}}{\pi_{1}e^{\varepsilon}+1-\pi_ {1}}\geq 0\), and so we can get \(\frac{dH_{\xi}(\varepsilon)}{d\varepsilon}\leq 0\). When \(\frac{1}{2}\leq\pi_{1}\leq 1\), we have \(2\pi_{1}-1\geq 0\) and \(\log_{2}\frac{e^{\varepsilon}(1-\pi_{1})+\pi_{1}}{\pi_{1}e^{\varepsilon}+1-\pi_ {1}}\leq 0\), and so we can obtain \(\frac{dH_{\xi}(\varepsilon)}{d\varepsilon}\leq 0\). The \(ENT\) of output of bivariate randomized response mechanism decreases as privacy budget \(\varepsilon\) increases. **Corollary 4**. : _The utility metric \(1-NE\) of output of bivariate randomized response mechanism increases as privacy budget \(\varepsilon\) increases._  
Proof. : According to Equation (38), the normalized entropy of output of bivariate randomized response mechanism is  
\[NE=H_{\xi}(\varepsilon)=-\frac{(1-\pi_{1})e^{\varepsilon}+\pi_{1}}{1+e^{ \varepsilon}}\log_{2}\frac{(1-\pi_{1})e^{\varepsilon}+\pi_{1}}{1+e^{ \varepsilon}}-\frac{(1-\pi_{1})+\pi_{1}e^{\varepsilon}}{1+e^{\varepsilon}} \log_{2}\frac{(1-\pi_{1})+\pi_{1}e^{\varepsilon}}{1+e^{\varepsilon}} \tag{42}\]  
Since \(\frac{dH_{\xi}(\varepsilon)}{d\varepsilon}\leq 0\), the normalized entropy \(NE\) of output of bivariate randomized response mechanism decreases as privacy budget \(\varepsilon\) increases. Therefore, the utility metric \(1-NE\) of output of bivariate randomized response mechanism increases as privacy budget \(\varepsilon\) increases. According to the Theorem 4, the entropy of output of bivariate randomized response mechanism decreases as privacy budget increases. The privacy-preserving level of output of bivariate randomized response mechanism decreases as privacy budget increases. By Corollary 4, the utility metric of \(1-NE\) of output of bivariate randomized response mechanism increases as privacy budget increases. Thus, the data utility using bivariate randomized response mechanism increases as privacy budget increases. According to Definition 8, Theorem 4 demonstrates the privacy-preserving monotonicity of bivariate randomized response mechanism. Privacy-preserving monotonicity shows the trade-off between privacy and utility based on Theorem 4 and Corollary 4. We introduce the following Chebyshev inequality [65] to prove the privacy-preserving monotonicity of multivariate randomized response mechanism. **Proof.