# A UAV-Assisted Stackelberg Game Model for Securing IoMT Healthcare Networks
## 3 System Model
### Proposed Stackelberg Security-Based Game Theory Algorithm for Securing IoMT Networks Using UAVs
#### 3.2.1 Stackelberg Game Theory Scheme  
The proposed scheme is based medical healthcare system where the information of each user is confidential; therefore, it is necessary to ensure the data's safety. Initially, we have considered 3D coordinates for the coverage area of UAVs. We have utilized the Euclidean distance formula for 3D coordinates in Equation (1) to calculate the distance between the UAV and end devices. \[D=\sqrt{\left(x^{2}{}_{UAV}-x^{2}{}_{IoT}\right)+\ \ \ \left(y^{2}{}_{UAV}-y^{2}{}_{IoT} \right)+\ \ \left(z^{2}{}_{UAV}-z^{2}{}_{IoT}\right)} \tag{1}\]  
where \(D\) is the distance between _UAV_ and IoMT device, and \(x,y\) and \(z\) are their coordinates. After finding the distance, we have introduced a game theory model which plays a vital role in many research areas, particularly for the security and safety of wireless networks. We have utilized the Stackelberg game theory model for IoMT medical healthcare networks by considering special features of game theory. The proposed scheme consists of three main parts. (1). The number of resources \(R=\{r_{1},r_{2},r_{3},\ldots.r_{n}\}\). (2). The set strategies \(S=\{s1,s2,s3,\ldots.\ldots.sn\}\). (3) Utility functions \(U=\{u1,u2,u3,\ldots.\ldots.un\}\). In wireless networks, each user is considered a player; in this scenario, we have considered the Stackelberg game theory model for security purposes. This model described the competition between two users: attacker (leader) and defender (follower). Initially, the attacker may choose a target from Equation (2) where \(T\) is a set of targets (\(T\)). \[T=\{r1,r2,r3,\ldots.\ldots rn\} \tag{2}\]  
The victim or defender (UAV) will defend its resource from the attacker node; the set of victim resources (\(Rs\)) is described in Equation (3):  
\[Rs=\{r1,r2,r3,\ldots.\ldots rn\} \tag{3}\]  
Assume a target \(t_{i}\) is attacked and recovered; in this case, the defender's utility function is \(U_{d}^{c}(t_{i})\) (as mentioned in Equation (4). If it is not recovered \(U_{d}^{u}(t_{i})\) whereas the attacker's utility function \(U_{d}^{c}(t_{i})\) is greater in this case the utility value will be negative for the victim side as indicated in Equation (5). \[U_{d}^{c}(t_{i})-U_{d}^{u}(t_{i})\ >\ 0 \tag{4}\]  
\[U_{d}^{u}(t_{i})-U_{d}^{c}(t_{i})\ >\ 0 \tag{5}\]  
From Equation (3), we can observe that if the target is recovered, the victim's or defender's utility function is positive. In case the attacker successfully hacks the data, the utility function of a victim is negative, as shown in Equation (4). Usually, in security applications, the attacker's pure strategy is based on a set of specific targets. In contrast, in the case of a mixed strategy, the attacker's strategy is based on the probability of attacking targets as vectors A= [a_1, a_2,......a_n]. The victim's pure strategy is to protect the targets of vector \(\in\) {0,1} n. Let us assume that D \(\in\) {0,1} n the set of possible coverage vectors, and \(m\) the vector of coverage probabilities. The defender containsthe set of the mixed strategies \(M\) is defined as the vector of probabilities of choosing each \(d\in\mathrm{D}\). For strategy \(M\), the defender's utility is defined in Equation (6):  
\[U_{d}(M,a)=\sum_{i=1}^{n}a_{i}(p_{i}U_{d}^{m}(t_{i})+(1-m_{i})U_{d}^{u}(t_{i})) \tag{6}\]  
The attacker's utility function based on mixed strategy where the attacker targets the particular resource from the set of coverage probabilities can be calculated from Equation (7). \[U_{a}(M,a)=\sum_{i=1}^{n}a_{i}(m_{i}U_{d}^{m}(t_{i})+(1-m_{i})U_{d}^{u}(t_{i})) \tag{7}\]  
The Nash equilibrium may be calculated in games using symmetric security. In these instances, the defender will choose a strategy (\(M\)) that maximizes their utility above any alternative strategy (\(M^{\prime}\)) available in Equation (8). \[U_{d}(M,a)>U_{d}(M^{\prime},a) \tag{8}\]  
In the above equation \(U_{d}(M,a)\) is the best strategy from all possible strategies and hence offers higher utility value.