## 2 Attack Position Assessment
### Calculation of Attack Position  
The defender cannot accurately understand the attacker's position, which constitutes an incomplete information game. This paper employs the Harsanyi transformation method to convert the game into a complete but imperfect information game for analysis [17]. The defender is aware that the attacker has a probability \(p\) of targeting the UC and a probability \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(dcp\) & \(dc\) & \(dp\) & \(nd\) \\ \hline \(a\) & \((1-2\alpha)\)\(S-Cx\), & \((1-2\alpha)\)\(S-Cx\), & \((1-2\alpha)\)\(S-Cx\), & \(S-Cx\), \\ \((2\alpha-1)\)\(S-Cdcp\) & \((2\alpha-1)\)\(S-Cdc\) & \((2\alpha-1)\)\(S-Cdp\) & \(-S\) \\ \(na\) & \(0,-\beta Cdcp\) & \(0,-\beta Cdc\) & \(0,-\beta Cdp\) & \(0,0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Payoff matrix. Let \(\max(U(dx_{m}))>U(nd)\), brought into Equations (6) and (7), have  
\[p>\frac{\beta Cdx_{m}}{2\alpha S-(1-\beta)Cdx_{m}}. \tag{8}\]  
The defender's dominant strategy is \(dx_{m}\). According to Tables 1 and 2, when \(Cas>(1-2\alpha)S>Cau\), the attacker's dominant strategy is \((a,na)\). So, \(((a,na),dx_{m},p)\) is a pure-strategy Bayesian-Nash equilibrium. Let \(\max(U(dx_{m}))\leq U(nd)\), brought into Equations (6) and (7), have  
\[p\leq\frac{\beta Cdx_{m}}{2\alpha S-(1-\beta)Cdx_{m}} \tag{9}\]  
The defender's dominant strategy is \(nd\). Due to \(\alpha>0\), according to Tables 1 and 2, the attacker's dominant strategy is \((a,a)\). So \(((a,na),dx_{m},p)\) is not a pure-strategy Bayesian-Nash equilibrium. Next, we will discuss mixed-strategy games. Let the attacker adopt attack strategies \(a\) and \(na\) with probabilities \(p_{a}\) and \((1-p_{a})\), respectively, while the defender employs defense strategies \(dx\) and \(nd\) with probabilities \(p_{dx}\) and \((1-p_{dx})\), respectively; \(p_{dx}+p_{nd}=1\). When attacker \(N_{a}\) uses strategy \((a,na)\), the expected return for each attack strategy is as follows:  
\[U(a)=pS(1-2\alpha p_{dx})-pCau \tag{10}\]  
\[U(na)=0 \tag{11}\]  
Let \(U(na)=U(a)\) be brought into Equations (10) and (11), as follows:  
\[p_{dx}^{*}=\frac{S-Cau}{2\alpha S} \tag{12}\]  
When attacker \(N_{a}\) uses strategy \((a,na)\), the expected returns for each defense strategy are as follows:  
\[U(dx)=(2\alpha-1)pp_{a}S-(1-\beta)pp_{a}Cdx-\beta Cdx \tag{13}\]  
\[U(nd)=-pp_{a}S \tag{14}\]  
Let \(U(dx)=U(nd)\) be brought into Equations (13) and (14), as follows:  
\[p_{a}^{*}=\frac{\beta Cdx}{2\alpha pS-(1-\beta)pCdx} \tag{15}\]  
According to the definition of the mixed-strategy Nash equilibrium, the attacker targets the UC with probability \(p_{a}^{*}\), while the defender responds with probability \(p_{dx}^{*}\). The mixed-strategy Nash equilibrium occurs when the chosen defensive strategy \(dx_{m}\) has the lowest cost. \(\square\)  
(3) When \(p<\frac{2\alpha S-Cdx_{m}}{2\alpha S-(1-\beta)Cdx_{m}}\) and \(Cas<(1-2\alpha)S<Cau\), there exists a pure strategy \(((na,a),dx_{m},p)\) as a Bayesian-Nash equilibrium; when \(p\geq\frac{2\alpha S-Cdx_{m}}{2\alpha S-(1-\beta)Cdx_{m}}\), there exists a mixed-strategy Bayesian-Nash equilibrium, with the attack position being the TSM. Proof. : When attacker \(N_{a}\) uses pure strategy \((a,na)\),  
\[U(dx)=p(-\beta Cdx)+(1-p)((2\alpha-1)S-Cdx) \tag{16}\]  
\[p>\frac{\beta Cdx_{m}}{2\alpha S-(1-\beta)Cdx_{m}}. \tag{8}\]0Let \(\max(U(dx_{m}))>U(nd)\) be brought into Equations (16) and (17), as follows:  
\[p>\frac{\beta Cdx_{m}}{2\alpha S-(1-\beta)Cdx_{m}}. \tag{8}\]1  
The defender's dominant strategy is \(dx_{m}\). According to Tables 1 and 2, when \(Cas<(1-2\alpha)S<Cau\), the attacker's dominant strategy is \((na,a)\). So, \(((na,a),dx_{m},p)\) is a pure-strategy Bayesian-Nash equilibrium. Let \(\max(U(dx_{m}))\leq U(nd)\), brought into Equations (16) and (17), have  
\[p>\frac{\beta Cdx_{m}}{2\alpha S-(1-\beta)Cdx_{m}}. \tag{8}\]2  
The defender's dominant strategy is \(nd\). Due to \(\alpha>0\), according to Tables 1 and 2, the attacker's dominant strategy is \((a,a)\).