## Appendix B Proof of Proposition 2. (i) Consider the insider strategies \(R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\) and \(R_{cyb}^{M}\ R_{\text{aut}}^{L}\) to respond to the defender strategy \(D_{\text{accept}}\.\)  
Let us compare \(U_{1}\big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{accept}}\ \big{)}\) with \(U_{1}\big{(}R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{accept}}\ \big{)}\). If \(C_{cyb}<2(p_{1}-p_{2})\beta\), it can be seen for \(\delta>0\) that  
\[U_{1}\big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{accept}}\ \big{)}<U_{1} \Big{(}R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{accept}}\ \Big{)} \tag{11}\]  
Also, consider the insider strategies \(R_{aut}^{M}\ R_{aut}^{L}\) and \(R_{cyb}^{M}\ R_{aut}^{L}\) to respond to the defender strategy \(D_{\text{reject}}\.\) Now, compare \(U_{1}\big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \big{)}\) with \(U_{1}\big{(}R_{cyb}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \big{)}\). For \(\delta>0\), it can be observed that  
\[U_{1}\Big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \Big{)}>U_{1} \Big{(}R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \Big{)} \tag{12}\]  
Again, consider the defender's strategies \(D_{\text{accept}}\ \) and \(D_{\text{reject}}\ \) to respond to the insider's strategy \(R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\.\) It can be deduced that if \(\delta\in\Delta_{1}\), where  
\[\Delta_{1}=\bigg{\{}\delta\in[0,1]\bigg{|}\delta>\frac{1}{1+\omega(1-2p_{1})} \ \bigg{\}} \tag{13}\]  
then  
\[U_{2}\Big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{accept}}\ \Big{)}<U_{2} \Big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \Big{)} \tag{14}\]  
Furthermore, consider the defender's strategies \(D_{\text{accept}}\ \) and \(D_{\text{reject}}\ \) to respond to the insider's strategy \(R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\.\) We have the following: If \(\delta\in\Delta_{2}\), where  
\[\Delta_{2}=\bigg{\{}\delta\in[0,1]\bigg{|}\delta>\frac{1}{1+\omega(1-2p_{2})} \ \bigg{\}} \tag{15}\]  
then  
\[U_{2}\Big{(}R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{accept}}\ \Big{)}<U_{2} \Big{(}R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \Big{)}. \tag{16}\]  
Since \(p_{1}>p_{2}\), from Equations (13) and (15), it can be noticed that \(\Delta_{1}\subset\Delta_{2}\) and, therefore, Equation (16) will also hold for \(\delta\in\Delta_{1}\), i.e., \(\delta>\frac{1}{1+\omega(1-2p_{1})}\). This shows that the profile of strategy \(\big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \big{)}\) is the unique pure-strategy Nash Equilibrium (NE) if \(\delta>\frac{1}{1+\omega(1-2p_{1})}\). Let \(\Delta_{3}=\Delta_{1}^{C}\smallsetminus\Big{\{}\delta\in[0,1]\Big{|}\delta=\frac{ 1}{1+\omega(1-2p_{1})}\ \Big{\}}\) and \(\Delta_{4}=\Delta_{2}^{C}\smallsetminus\Big{\{}\delta\in[0,1]\Big{|}\delta=\frac{ 1}{1+\omega(1-2p_{2})}\ \Big{\}}\). Thus, \(\Delta_{4}\subset\Delta_{3}\). Denote \(x^{*}=\frac{1}{1+\omega(1-2p_{1})}\) and \(y^{*}=\frac{1}{1+\omega(1-2p_{2})}\). Since \(p_{1}>p_{2}\), it is clear that \(x^{*}>y^{*}\). From Equations (A5) and (A6), if \(\delta<x^{*}\), i.e., \(\delta\in\Delta_{3}\), then  
\[U_{2}\Big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{accept}}\ \Big{)}>U_{2} \Big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \Big{)}\] (A9)  
In addition, if \(\delta<y^{*}\), i.e., \(\delta\in\Delta_{4}\), then  
\[U_{2}\Big{(}R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{accept}}\ \Big{)}>U_{2} \Big{(}R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \Big{)}.\] (A10)  
As \(\Delta_{4}\subset\Delta_{3}\), Equation (A9) also holds for \(\delta\in\Delta_{4}\). Therefore, if \(\delta<y^{*}\), then the profile of strategy \(\Big{(}R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{accept}}\ \Big{)}\) is the unique pure-strategy BNE of the game. Again, consider Equations (A8) and (A9). We know that if \(\delta\in\Delta_{2}\) and \(\delta\in\Delta_{3}\), i.e., \(y^{*}<\delta<x^{*}\), then Equations (A8) and (A9) hold.