# Risk Assessment of UAV Cyber Range Based on Bayesian-Nash Equilibrium
## 3 Application of Attack-Defense Tree Model in UAV Cyber Range
### Establish UAV Range Defense Model Based on Bayesian Static Game  
The Bayesian-Nash equilibrium is constructed by using the attack-defense tree model \(\text{AGM}=(A,S,D)\), where \(A=\{a_{1},a_{2},\cdots a_{6}\}\) represents the set of atomic attacks of the attacker, and the attack income is represented by the value after atomic attacks. Each attack path consists of multiple atomic attacks; \(S=\{s_{1},s_{2},\cdots s_{8}\}\) represents a status node; \(D=\{d_{1},d_{2},d_{3}\}\) represents a collection of defense behaviors, represented by the name of the defense behavior and the defense income. Static Bayesian game is a static game model with incomplete information, which represents a game in which participants act simultaneously with incomplete information. In this paper, a static Bayesian offensive and defensive game model is constructed that conforms to the actual offensive and defensive environment of the UAV cyber range. The participants \(UAV-ADM=((N_{A},N_{D}),(T_{A},T_{D}),(B_{A},B_{D}),(P_{A},P_{D}),(U_{A},U_{D}))\) include attackers and defenders, that is, the number of participants \(n=2\). The static Bayesian offensive and defensive game model for the offensive and defensive environment of UAV cyber range can be expressed as a quintuple, where:  
1. \(N=(N_{A},N_{D})\) represents the set of participants in the offensive and defensive game model. \(N_{A}\) and \(N_{D}\) represent attackers and defenders in the UAV range, respectively. 2. \(T=(T_{A},T_{D})\) represents the type set of participants in the offensive and defensive game model. Where \(T_{A}=\left(t_{1}^{A},t_{2}^{A},\cdots,t_{n1}^{A}\right)\) represents the set of types of attackers \(N_{A}\), such as internal attacks, external attacks, and spies; \(T_{D}=\left(t_{1}^{D},t_{2}^{D},\cdots,t_{n2}^{D}\right)\) represents a set of types of defenders \(N_{D}\), such as security experts, security analysts, and security operations personnel. In the actual UAV range, one side of the offense and defense does not fully understand the benefits of the other side. Therefore, when analyzing the static Bayesian offensive and defensive game model, the method of \"Harsanyitransformation\" is used to assume a fictitious player, and the fictitious player first decides the different types of other players, and replaces the uncertainty of information with the uncertainty of type. 3. \(B=(B_{A},B_{D})\) represents the action set of players in the game model. It represents a collection of actions by different types of attackers \(N_{A}\), such as DoS, backdoors, fuzzers; and a set of actions \(B_{D}=\left(b_{1}^{D},b_{2}^{D},\cdots,b_{m2}^{D}\right)\) that represent different types of defenders, such as authentication, transmission encryption, and intrusion detection. 4. \(P=(P_{A},P_{D})\) represents the prior probability set of players in the offensive and defensive game model. Where \(P_{A}=P(t_{D}|t_{A})\) represents the probability that the attacker judges the type \(t_{A}\) of the defender in the case of type \(t_{D}\); \(P_{D}=P(t_{A}|t_{D})\) indicates the probability that the defender determines the type \(t_{D}\) of the attacker in the case of type \(t_{A}\). 5. \(U=(U_{A},U_{D})\) represents the income function set of participants in the offensive and defensive game model. \(\forall b_{A}\in B_{A},b_{D}\in B_{D},t_{A}\in T_{A},t_{D}\in T_{D}\), \(U_{A}(b_{A},b_{D},t_{A})\) represents the revenue function of the attacker when the attacker is of the type \(t_{A}\), the attacker takes action \(b_{A}\), and the defender takes action \(b_{D}\); \(U_{D}(b_{A},b_{D},t_{D})\) represents the revenue function of the defender when the defender is of the type \(t_{D}\), when the attacker takes action \(b_{A}\), and when the defender takes action \(b_{D}\). According to the above, in order to better analyze the attack and defense game model of the UAV cyber range, the quantitative calculation method of attack and defense benefits is given.