# Deployment Optimization of Defense Stations in an Attack-Defense Aerial War Game
## 3 Simulation Results and Statistical Analysis
The average hit on the south post (0.33) is the lowest among all the four cases, because two attackers are hit by two interceptors fired from nearby defense stations, respectively, the interceptors originally chasing these two attackers are spared to pursue the other attackers aiming for the south post. The average hit on the north post (1.72) is moderate compared with the other cases while no defense station is placed within 40 km of the former. The defense station loss rate (55.7%) is the highest among all the four cases because two out of three defense stations that have no cohort station within 40 km of them and become easy targets when all their interceptors are exhausted. Some attackers originally aiming for the two hit defense stations detour to attack nearby defense stations. One of the nearby defense stations, without cover from other cohort stations within 40 km, already fired all its interceptors and becomes a sitting duck. The other nearby defense station still has some interceptors available, but one of the approaching attackers has already been marked by two interceptors. Abiding by the game rules, this defense station will not fire another interceptor against the approaching attacker, and is hit before the chasing interceptors can catch the latter. One defense station \begin{table}
\begin{tabular}{c c c c c c} \hline
**Defense Cost Function** & \(\zeta_{1}\) & \(\zeta_{2}\) & \(\zeta_{3}\) & \(\zeta_{4}\) & **Empirical** \\ \hline average hits on major post & 1.81 & 0.93 & 1.03 & 0.9 & 2.67 \\ \hline average hits on north post & 0.09 & 1.85 & 1.72 & 2.71 & 0.56 \\ \hline average hits on south post & 1.23 & 1.8 & 0.33 & 0.77 & 1.24 \\ \hline defense stations lost (\%) & 48.3 & 29.2 & 55.7 & 41.8 & 54 \\ \hline interceptors launched (\%) & 82.9 & 84.113 & 80.488 & 80.75 & 90.763 \\ \hline attackers lost (\%) & 79.95 & 78.275 & 75.8 & 75.875 & 74.2 \\ \hline average defense cost & 4.762 & 1.305 & 5.482 & 6.022 & - \\ \hline median of defense cost & 4.6 & 1.344 & 5.33 & 5.8 & - \\ \hline
80\% confidence interval & 3.618 & 2.1163 & 3.8185 & 3.5395 & - \\ \hline
95\% confidence interval & 6.64 & 3.171 & 6.04 & 6 & - \\ \hline \end{tabular}
\end{table} 
Table 5: Statistics of loss against pincer attack formation. is almost guaranteed to be hit by an attacker, which originally aimed for the major post and later detours to strike the former. The attacker loss rate (75.8%) is comparable to the other cases, which is related to the high hit number on the north post and the large number of lost defense stations. With the optimal deployment plan based on \(\zeta_{4}\), the average hit on the major post (0.9) is the lowest among all the four cases since four defense stations are placed within 40 km of the former. The average hit on the south post (0.77) is relatively low, comparable to the low hit on the north post based on \(\zeta_{1}\), under slant attack formation. However, the average hit on the north post (2.71) is the highest among all the four cases since no defense station is placed around it. The attacker loss rate (75.875%) is comparable to other cases, which is related to the high hit number on the north post. The 80% confidence interval lies between 2.1 and 3.9, and the 95% confidence interval lies between 3.1 and 6.7. Both values are higher than their counterparts under slant attack formation because the south attack stations are closer to the defense force, making the latter more difficult to protect. Figure 10a shows that the CDF curves of impact score (hit) on the major post based on \(\zeta_{1}\) and \(\zeta_{4}\) are similar. The CDF curve on \(\zeta_{2}\) shows fewer games with score 0 but more games with score 1, compared with those on \(\zeta_{1}\) and \(\zeta_{4}\). The CDF on \(\zeta_{1}\) shows score 0 in 44 games, score 1 in 44 games, and the highest score is 3. The CDF on \(\zeta_{2}\) shows score 0 in 2 games, score 1 in 77 games, and the highest score is 5. The CDF on \(\zeta_{3}\) shows score 0 in 88 games and the highest score is 2.