# Deployment Optimization of Defense Stations in an Attack-Defense Aerial War Game
## 1 Introduction  
War games have been widely used to gain insights and lessons from a wide variety of scenarios that are otherwise improbable to observe without playing the games. Useful information like critical factors and effective strategy manifested in the games can help reduce the risk in real lives. In this work, an attack-defense aerial war game is developed to maximize the capability of defense force in protecting a target area from an attack force and gain some insights on the nature of the games. A typical aerial war game is staged in a specific scenario and proceed in various constituent episodes. The players in a game try to achieve their own goal by using specific methods or techniques, and follow predefined game rules in conducting each episode. For example, in [1], a cooperative combat between two teams of unmanned aerial vehicles (UAVs) was simulated. Three training techniques were proposed to enhance the performance of a multi-agent deep Q-learning (MADQN) method. A scenario-transfer training was exercised in a simple task, a self-play training was used to iteratively improve the performance of UAVs in a team, and a rule-coupled training was embedded in the combat rules to constrain the exploration space. However, some practical factors such as kill probability and maximum flying distance were not considered. In [2], an air combat between two UAVs was modeled as an optimization problem, and a moving-time horizon scheme was proposed to search for the optimal trajectories of UAVs. In [3], two opponent UAVs competed to fire missiles against each other by applying a deep learning method to choose an optimal maneuver angle, based on the predicted state and a decision objective function. The proposed method was claimed to be more flexible in decision-making and more economic in computational time than some previous works. Similar conflicts between two forces of same size and composition were also studied [1; 2; 3]. In this work, an aerial war game is developed between an attack force and a defense force of comparable capabilities. There are many episodes that involve three players. For example, in [4], a (TAD) game was developed, with an attacker pursuing a moving target while evading a defender. The moving target cooperated with the defender, and the attacker switched role between pursuer and evader. In [5], an active target defense differential game (ATDDG) was proposed, in which a missile (attacker) tracked an aircraft (target) and the latter fired a defense missile (interceptor) against the former. A closed-form solution of the ATDDG was implemented as a state-feedback controller for the target and the interceptor, aiming to increase the opportunity of target survival. The attacker and the interceptor flew at the same speed, the guidance rules of the attacker were unknown, and the escape scheme of attacker was not mentioned. In [6], an active target-attacker-defender (ATAD) differential game involved an attacker, multiple targets, and a defender. The attacker aimed at the closest target, and the defender protected the targets or intercepted the attacker. The interaction between agents was modeled as a linear quadratic differential game (LQDG), reaching an open-loop Nash equilibrium. The agents could change their strategies by taking a receding horizon approach to compute their trajectories accordingly. Interception is a common practice to protect targets from attackers [4; 5; 6], which is also incorporated in our aerial war game. Some coordination is required when multiple players are involved in a game scenario. For example, in [7], multiple fighters were dispatched to suppress an integrated air defense system (IADS). An improved chaotic particle swarm optimization (I-CPSO) algorithm was applied on both sides. The simulations included consumption of air-to-air missiles, distance to non-escape zone and jamming distance, but the efficacy of the dispatch plan was not verified. In [8], a beyond-visual-range (BVR) air combat between two fleets of UAVs was formulated as a cooperative decision-making problem.