# Privacy-Preserving Monotonicity of Differential Privacy Mechanisms
## 7 Numerical Results
In Figure 10, the utility metric \(1-NE\) of data using bivariate randomized response mechanism increases as privacy budget increases. This verifies Corollary 4. In Figure 11, we observe that \(H_{\xi}(\pi_{1})\) (Equation (38)) monotonically increases as \(\pi_{1}\) increases to bivariate randomized response mechanism, when \(0\leq\pi_{1}\leq\frac{1}{2}\). When \(\frac{1}{2}\leq\pi_{1}\leq 1\), \(H_{\xi}(\pi_{1})\) is monotonically decreasing as \(\pi_{1}\) increases. Thus, \(H_{\xi}(\pi_{1})\) has the maximal value \(1\) when \(\pi_{1}=\frac{1}{2}\). \(H_{\xi}(\pi_{1})\) is symmetry on \(\pi_{1}=\frac{1}{2}\). That is to say, \(H_{\xi}(\pi_{1})=H_{\xi}(1-\pi_{1})\) for all \(\pi_{1}\in[0,1]\). This reason is \(P_{0}(\pi_{1})=P_{1}(1-\pi_{0})\) and \(P_{1}(\pi_{1})=P_{0}(1-\pi_{0})\). In Figure 10, the utility metric \(1-NE\) of data using bivariate randomized response mechanism is equal to \(0\) when \(\pi_{1}=\frac{1}{2}\). Thus, this results in the worst availability of data and gets the best privacy preservation. Figure 6: Modulus of characteristic function of query results of Gaussian mechanism. Figure 7: Entropy of output of exponential mechanism. Figure 8: Utility metric \(1-NE\) of output of exponential mechanism. Figure 10: Utility metric \(1-NE\) of output of bivariate randomized response mechanism. Figure 6.0 Entropy of output of bivariate randomized response mechanism. For privacy-preserving monotonicity of multivariate randomized response mechanism, we can also see that entropy of output decreases as privacy budget \(\varepsilon\) increases from Figure 6.1 This is consistent with Theorem 5. We can obtain that the utility metric \(1-NE\) of data using multivariate randomized response mechanism increases as privacy budget increases from Figure 6.2 This demonstrates Corollary 5. We can conclude that entropy of multivariate randomized response mechanism is maximal value \(\log_{2}n\) when \(\pi_{1}=\pi_{2}=\ldots=\pi_{n}=\frac{1}{n}\) from Figure 6.3 At the same time, the utility metric \(1-NE\) of data using multivariate randomized response mechanism is equal to 0 in Figure 13, which results in the worst availability of data and gets the best privacy preservation. We observe that the entropy of output using Laplace mechanism deceases as privacy budget \(\varepsilon\) increases from Figure 14, which conforms to Theorem 6. In Figure 15, the entropy of output using Gaussian mechanism decreases as privacy budget \(\varepsilon\) increases. The result further demonstrates Theorem 7. For Laplace mechanism and Gaussian mechanism, we can see that the entropy increases as sensitivity increases from Figures 14 and 15. Moreover, we also get that the entropy increases as sensitivity increases to Laplace mechanism and Gaussian mechanism from Equations (51) and (52). Conversely, the availability of data using Laplace mechanism and Gaussian mechanism decreases as sensitivity increases based on Equations (20) and (25). Figure 6.4 Entropy of output of bivariate randomized response mechanism under different \(\pi_{1}\). Figure 6.5 Entropy of output of multivariate randomized response mechanism. For the research and applications of differential privacy, we can theoretically and numerically show the privacy-preserving monotonicity of privacy metrics of the differential privacy mechanisms to measure the trade-off between privacy and utility. According to the theoretical and numerical results, the privacy-preserving monotonicity of privacy metrics is not a good approximation of the privacy-preserving monotonicity of privacy metrics. Figure 6.6 Entropy of output of Laplace mechanism. Figure 6.7 Utility metric \(1-NE\) of output of multivariate randomized response mechanism. Figure 6.8 Entropy of output of Gaussian mechanism.