# Deployment Optimization of Defense Stations in an Attack-Defense Aerial War Game
## 5 Scenarios with Perfect Kill Probability
Figure 19a shows an optimal deployment plan of defense stations against pincer attack formation, with kill probability set to 1, based on defense cost \(\zeta_{1}\). There are six defense  
Figure 17: Convergence of defense cost against pincer formation with kill probability set to 1, black: \(\zeta_{1}\), orange: \(\zeta_{2}\), green: \(\zeta_{3}\), purple: \(\zeta_{4}\). Figure 18: Optimal deployment plan of defense stations against slant attack formation, with kill probability set to 1, based on (**a**) \(\zeta_{1}\); (**b**) \(\zeta_{2}\); (**c**) \(\zeta_{3}\); (**d**) \(\zeta_{4}\); square with cross: major post, square with slash: minor post, blue square: defense station. stations placed within 40 km of the major post, one defense station is within 40 km of the north post, and two defense stations are near the south post, but not within 40 km of the latter. One defense station is placed between the major post and the north post to cover both. More defense stations are placed around the major post as compared to the counterpart case under slant attack formation, because the south attack stations are closer to the major post now. Figure 19b shows an optimal deployment plan of defense stations based on defense cost \(\zeta_{2}\). There are five defense stations placed near the major post, with four within 40 km of the latter, two defense stations are placed near the north post, with one to the north and within 40 km of the latter. No defense station is placed within 40 km of the south post, three defense stations are placed between the major post and the south post to cover both. The number of defense stations near the major post is compatible with the case on \(\zeta_{1}\). Figure 19c shows an optimal deployment plan of defense stations based on defense cost \(\zeta_{3}\). There are six defense stations placed near the major post, with three within 40 km of the latter, two and one defense stations are placed near the north post and the south post, respectively, but none of them are within 40 km of either post, and one defense station is placed between the major post and the south post to cover both. Although the weighting of a minor post is on par with the major post, the major post is still covered with many more defense stations because it is targeted by many more attackers than a minor post. Figure 19d shows an optimal deployment plan of defense stations based on defense cost \(\zeta_{4}\). There are six defense stations placed near the major post, with five within 40 km of the latter, two defense stations are placed near the north post, with one within 40 km of the latter, one defense station is placed to the south of the south post and within 40 km of the latter, and one defense station is placed between the major post and the south post to cover both. As was mentioned in the beginning of this section, the deployment plan of defense stations determines the script and outcome of any game if the kill probability is set to 1. With the deployment plans shown in Figure 18, the impact score on the major post is 0 on all the four types of defense cost function. The scores on the south post are 0, 0, 0, and 1 on \(\zeta_{1}\), \(\zeta_{2}\), \(\zeta_{3}\), and \(\zeta_{4}\), respectively. The score of 1 on \(\zeta_{4}\) is made by one attacker which successfully evades two interceptors, with one of them fired from a defense station within 40 km of the south post. The attacker finds at most one intercept point on route. The scores on the north post are 1, 0, 0, and 1 on \(\zeta_{1}\), \(\zeta_{2}\), \(\zeta_{3}\), and \(\zeta_{4}\), respectively. The score of 1 on \(\zeta_{4}\) is taken because no defense station is placed within 40 km of the north post.