# Privacy-Preserving Monotonicity of Differential Privacy Mechanisms
## 7 Numerical Results  
In this section, we conduct numerical experiments for these conclusions of Section 6. We conduct these numerical experiments by implementing them with MATLAB (R2013b) and run our experiments on a desktop computer with Intel i5-2400 3.10 GHz processor, 4 GB RAM, and Windows 7 platform. The detailed parameter settings are shown in Table 2.  
In Figure 3, we can observe that the expected estimation error decreases as privacy budget \(\epsilon\) increases when using Laplace mechanism for numeric data. This is consistent with Theorem 1. Figure 4 shows that the modulus of characteristic function of query results of Laplace mechanism increases as privacy budget increases. This verifies Corollary 1. In Figure 5, we know that the expected estimation error of query results using Gaussian mechanism decreases as privacy budget \(\epsilon\) increases, which is consistent with Theorem 2. Conversely, we observe that the modulus of characteristic function of query results of Gaussian mechanism increases as privacy budget increases from Figure 6. This further demonstrates Corollary 2. We see that the expected estimation error increases as sensitivity increases to Laplace mechanism and Gaussian mechanism from Figures 3 and 5. We can also conclude that expected estimation error increases as sensitivity increases to Laplace mechanism and Gaussian mechanism from Equations (18) and (23). Conversely, we observe that the modulus of characteristic function of query results using Laplace mechanism and Gaussian mechanism decreases as sensitivity increases from Figures 4 and 6. We can also conclude that the modulus of characteristic function of query results using Laplace mechanism and Gaussian mechanism decreases as sensitivity increases from Equations (20) and (25). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Mechanism** & **Parameter** & **Value** \\ \hline Laplace mechanism & \(\Delta f\) & \(\Delta f=1\), \(\Delta f=2\), \(\Delta f=3\) \\  & \(t\) & \(t=1\) \\ Gaussian mechanism & \(\Delta_{2}f\) & \(\Delta_{2}f=1\), \(\Delta_{2}f=2\), \(\Delta_{2}f=3\) \\  & \(t\) & \(t=1\) \\  & \(u_{1}=2\), \(u_{2}=2\), \(u_{3}=2\), \(u_{4}=2\), \(u_{5}=2\) \\ Exponential mechanism & \(u_{1}\), \(u_{2}\), \(u_{3}\), \(u_{4}\), \(u_{5}\) & \(u_{1}=1\), \(u_{2}=3\), \(u_{3}=2\), \(u_{4}=3\), \(u_{5}=1\) \\  & \(u_{1}=1\), \(u_{2}=4\), \(u_{3}=3\), \(u_{4}=1\), \(u_{5}=1\) \\ Bivariate randomized response & \(\pi_{1}\) & \(\pi_{1}=0.5\), \(\pi_{1}=0.7\), \(\pi_{1}=0.9\) \\  & \(\epsilon\) & \(\varepsilon=0.1\), \(\epsilon=0.2\), \(\epsilon=0.3\) \\  & & \(\pi_{1}=0.25\), \(\pi_{2}=0.25\), \(\pi_{3}=0.25\), \(\pi_{4}=0.25\) \\ Multivariate randomized response & \(\pi_{1}\), \(\pi_{2}\), \(\pi_{3}\), \(\pi_{4}\) & \(\pi_{1}=0.1\), \(\pi_{2}=0.2\), \(\pi_{3}=0.3\), \(\pi_{4}=0.4\) \\  & & \(\pi_{1}=0.1\), \(\pi_{2}=0.4\), \(\pi_{3}=0.4\Figure 4: Modulus of characteristic function of query results of Laplace mechanism. Figure 5: Expected estimation error of query results of Gaussian mechanism. Figure 3: Expected estimation error of query results of Laplace mechanism. In Figure 7, we can get the monotonically decreasing curve of entropy of output using exponential mechanism. The result verifies Theorem 3. Figure 8 shows that the utility metric \(1-NE\) of data using exponent mechanism increases as privacy budget increases. This result is covered in Corollary 3. By Theorem 3, when \(\varepsilon=0\) or \(u_{1}(x,r_{1})=u_{2}(x,r_{2})=\ldots=u_{n}(x,r_{n})=0\), \(H(\varepsilon)\) has the maximal value \(\log_{2}n\). At the same time, all random variables have the maximum uncertainty. Thus, this leads to the maximum entropy and gets the best privacy preservation. At the same time, the utility metric \(1-NE\) is equal to \(0\) in Figure 8, which results in the worst availability of data. Entropy of output using bivariate randomized response mechanism decreases as privacy budget \(\varepsilon\) increases from Figure 9. This illustrates Theorem 4.