## 5 Numerical Experiments and Discussion
### Case Study-2  
The survival time payoff matrix \(\mathbf{H}\) containing the interval-based normalized survival times (values between (0, 1]) are given in Table 3. The normalized values in the range \([0.2,0.4]\) may indicate short (small) survival time, latency or energy consumption, \([0.5,0.7]\) indicates medium survival time, latency or energy consumption, and \([0.8,1]\) indicates long survival time, latency or energy consumption. For the routing mechanisms, Table 4 presents the energy \(\mathbf{p}\) and the latency \(\mathbf{q}\) values. Suitable units for survival time, energy, and latency can be obtained from the actual network. A discussion on obtaining the above data from actual WSN is presented at the end of this section. The case study has the following key structure: The defender can change the routing policy from time to time. The defender can pick from any six routing mechanisms, say: \(\mathbf{R1},\mathbf{R2},\ldots\mathbf{R6}\). On the other hand, the adversary can change the attacking mechanism from time to time. The adversary can pick from any six attack mechanisms, say: \(\mathbf{A1},\mathbf{A2},\ldots\mathbf{A6}\).  
The grid for \(\varepsilon_{1}\) is generated by taking 10 equally spaced intervals of \([0,0.2\times q_{max}]\), \([0,0.2\times\overline{q}_{max}]\), and \([0,0.2\times\underline{q}_{max}]\) for solving Formulations (12a-f)-(14a-f), respectively. Similarly, for \(\varepsilon_{2}\), the grid is generated by taking 10 equally spaced intervals of \([0,0.2\times p_{max}]\), \([0,0.2\times\overline{p}_{max}]\), and \([0,0.2\times p_{max}]\) for solving Formulations (12a-f)-(14a-f), respectively. All the models are solved using the GLPK solver ([[https://www.gnu.org/software/glpk/](https://www.gnu.org/software/glpk/)]([https://www.gnu.org/software/glpk/](https://www.gnu.org/software/glpk/)), accessed on 1 September 2022 ). The Pareto surface obtained after solving Formulation (12a-f) is depicted in Figure 1a, and the Pareto surfaces obtained after solving Formulations (13a-f) and (14a-f) are illustrated in blue and green colored surfaces in Figure 1b, respectively. Table 5 displays the results of the case study at \(\varepsilon_{1}=\varepsilon_{2}=0\), which indicates the minimax game values. From Table 5, it can be concluded that if the defender ignores the latency- and energy-related objectives, then playing \(\mathbf{R1},\ \mathbf{R2}\) or \(\mathbf{R6}\) equally will likely be the best for LPP. Practically, it means that the network administrator can alternate between \(\mathbf{R1},\ \mathbf{R2}\) or \(\mathbf{R6}\) to obtain the longest possible survival time, while preserving the privacy location of the sink node. Other operational points can be extracted from Figure 1b.  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & \(\mathbf{R1}\) & \(\mathbf{R2}\) & \(\mathbf{R3}\) & \(\mathbf{R4}\) & \(\mathbf{R5}\) & \(\mathbf{R6}\) \\ \hline \(\mathbf{p}\) & 0.9 & 0.7 & 0.2 & 0.6 & 0.5 & 0.3 \\ \hline \(\mathbf{q}\) & 0.4 & 0.6 & 0.7 & 0.3 & 0.8 & 0.5 \\ \hline \end{tabular}
\end{table}
Table 4: Energy (\(\mathbf{p}\)) and Latency (\(\mathbf{q}\)) for each routing mechanism in Case Study-2.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & \(\mathbf{A1}\) & \(\mathbf{A2}\) & \(\mathbf{A3}\) & \(\mathbf{A4}\) & \(\mathbf{A5}\) & \(\mathbf{A6}\) \\ \hline
**R1** & [0.5, 0.7] & [0.8, 1] & [0.5, 0.7] & [0.8, 1] & [0.2, 0.4] & [0.2, 0.4] \\ \hline
**R2** & [0.2, 0.4] & [0.5, 0.7]