## Appendix B Proof of Proposition 2.
This implies that there exists a mixed-strategy BNE for the game, which is computed as below:  
Suppose that the insider randomizes his/her strategies \(R_{aut}^{M}R_{aut}^{L}\) and \(R_{\text{cyb}}^{M}R_{aut}^{L}\) with the corresponding probabilities \(\lambda_{1}\) and \(1-\lambda_{1}\), respectively. Also, the defender randomizes his/her strategies \(D_{\text{accept}}\) and \(D_{\text{reject}}\) with the corresponding probabilities \(\eta_{1}\) and \(1-\eta_{1}\), respectively. Thus, the expected payoffs of the insider corresponding to his/her strategies are given by  
\[\mathbb{E}_{1}\Big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\ \Big{)}=\eta_{1}U_{1} \Big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{accept}}\ \Big{)}+(1-\eta_{1})U_{1} \Big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \Big{)}\] (A11)  
\[\mathbb{E}_{1}\Big{(}R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\ \Big{)}=\eta_{1}U_{1} \Big{(}R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{accept}}\ \Big{)}+(1-\eta_{1})U_{1} \Big{(}R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \Big{)}\] (A12)  
The insider must be indifferent to choosing between the strategies \(R_{aut}^{M}R_{aut}^{L}\) and \(R_{\text{cyb}}^{M}R_{\text{aut}}^{L}\) from the expected payoff given by Equations (A11) and (A12). Hence, we must have \(\mathbb{E}_{1}\big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\ \big{)}=\mathbb{E}_{1} \Big{(}R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\ \Big{)}\). On simplifying this equation, we obtain  
\[U_{1}\Big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \Big{)}>U_{1} \Big{(}R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \Big{)} \tag{12}\]0 (A13)  
Additionally, the expected payoffs of the defender corresponding to his/her strategies are given by  
\[U_{1}\Big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \Big{)}>U_{1} \Big{(}R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \Big{)} \tag{12}\]1 (A14)  
\[U_{1}\Big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \Big{)}>U_{1} \Big{(}R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \Big{)} \tag{12}\]2 (A15)  
The defender would be indifferent to choosing between his/her strategies \(D_{\text{accept}}\) and \(D_{\text{reject}}\) and, therefore, we must have \(\mathbb{E}_{2}\big{(}D_{\text{accept}}\ \big{)}=\mathbb{E}_{2}\big{(}D_{\text{reject}}\ \big{)}\). On simplifying this equation based on Equations (A14) and (A15), we obtain  
\[U_{1}\Big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \Big{)}>U_{1} \Big{(}R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \Big{)} \tag{12}\]3 (A16)  
Summarizing the above results, we obtain the solution of the game as given in Equation (3). (ii) If \(C_{\text{cyb}}>2\beta(p_{1}-p_{2})\), it can be observed that  
\[U_{1}\Big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \Big{)}>U_{1} \Big{(}R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{reject}}\ \Big{)} \tag{12}\]4 (A17)Furthermore, if \(\delta\in\Delta_{1}\) then the defender's strategy \(D_{\text{neject}}\) is the best response to the insider's strategy \(R_{\text{aut}}^{M}\)\(R_{\text{aut}}^{L}\). Thus, from Equations (A4) and (A17), it follows that the profile of strategy \(\left(R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{neject}}\ \right)\) is the unique pure-strategy BNE of the game. On the other hand, if \(\delta\in\Delta_{3}\), then it can be noticed that \(D_{\text{accept}}\) is the defender's best response to the insider's strategy \(R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\). Hence, the profile of strategy \(\left(R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{accept}}\ \right)\) is the unique pure-strategy BNE of the game. Also, it can be noticed that under the condition \(C_{\text{c}yb}>2\beta(p_{1}-p_{2})\), there is no mixed-strategy BNE. The above results can be summarized as presented in Equation (4). \(\square\)  
Appendix C Proof of Proposition 3. (i) \(\delta<x^{*}\), i.e., \(\delta\in\Delta_{3}\) and \(p_{2}<\frac{1}{2}\).