# Deployment Optimization of Defense Stations in an Attack-Defense Aerial War Game
## 1 Introduction
Both fleets competed to maximize the difference between predominance and threat in terms of relative distance and relative height. A zero-sum matrix game was developed by using a double oracle and neighborhood search (DO-NS) algorithm. Aerial war games were also played to improve the tactical operations. In [9], an optimization procedure was proposed to fire missiles against high-velocity attackers, which incorporated practical factors such as multiple phases (initial, midcourse, and terminal) of attackers, multiple targets, interceptor cost, interceptor kill probability, detection probability of attackers, and risk attitude (human psychological features under stress) of the defender. Combinatorial mathematics was applied to model multiple phases and multiple targets, and a cumulative prospect theory (CPT) was used to model the risk attitude of the defender. The assignment plan of interceptors was acquired, but its efficacy was not verified by simulations. In [10], a static multi-objective weapon-target assignment (SMWTA) algorithm was proposed to assess combat condition, based on the damage to opponent, cost of missiles, and loss of fighting capacity inflicted by an opponent counterattack. A non-dominated sorting genetic algorithm III (NSGA-III) was applied to search for Pareto solutions in the decision space. An assignment plan of fighters and missiles to targets was obtained, but its efficacy was not verified. The fighting capacity was counted in terms of the fighter value, the target value, and the launch range of missiles, but some practical factors such as escape capability of fighters and kill probability against moving object were not considered. Better assignment plan of agents can inflict more serious wreck on the enemy force [7; 8; 9; 10], which is also included in our work. Evading maneuvers has been implemented in aerial war games. For example, in [11], two homing missiles shared their measured line-of-sight (LoS) with each other to improve their tracking performance on an aerial target. A blinding-evasion guidance law was proposed to confuse the two homing missiles, and an evasive maneuver was taken to increase the target survivability. Defense on a border or a territory has been a popular theme of war games. In [12], a strategy based on dominant region was proposed for multiple defenders to intercept an intruder before it entered a target area. The game highlighted the approaching method of intruder and the interception method of defenders. The intruder maneuvered in response to the defenders' strategies instead of following a premeditated plan. In [13], a reach-avoidperimeter defense game was played between intruders and defenders. Multiple intruders were sent to the target area, and the defenders deployed around the target perimeter would strike the intruders with a cooperative defense strategy. In [14], a border defense scenario was studied, where two intruders sneaked as close to the border as possible and an UAV would try to capture these intruders as far away from the border as possible. A differential game was formed to search for the optimal motions of the UAV and the cooperative intruders, respectively. In [15], evolution strategies (ES) were applied to a team of fixed-wing planes in attacking an enemy base station while avoiding its guarding quadcopters. The strategies were also applied to two teams, equipped with the same number of planes and base stations, operating on different policies to defeat the opponent planes or attack the opponent base stations. The aerial war game in our work is staged to protect a specific area with multiple defense stations, similar to [12; 13; 14; 15], but the area size and the number of agents are increased to inspect some nuances aroused by the complicated game scenario. Maneuvering of attackers and defenders directly determines the outcome of their engagement.