## Appendix A  
**Proof of Proposition 1.** (i) If \(C_{\text{cvb}}<\beta p_{1}\), then we have \(U_{1}\big{(}R_{\text{aut}}^{M}R_{\text{aut}}^{L},D_{\text{accept}}\big{)}<U_{ 1}\Big{(}R_{\text{cyb}}^{M}R_{\text{aut}}^{L}\,,\,D_{\text{accept}}\big{)}\) and \(U_{1}\big{(}R_{\text{aut}}^{M}R_{\text{aut}}^{L}\,,D_{\text{reject}}\big{)}\)\(>U_{1}\Big{(}R_{\text{cyb}}^{M}R_{\text{aut}}^{L}\,,D_{\text{reject}}\Big{)}\). Additionally, if \(\delta>\frac{1}{1+\omega(1-p_{1})}\), then \(U_{2}\big{(}R_{\text{aut}}^{M}R_{\text{aut}}^{L}\,,D_{\text{accept}}\big{)}<U _{1}\big{(}R_{\text{aut}}^{M}R_{\text{aut}}^{L}\,,D_{\text{reject}}\big{)}\). Let \(x_{1}=\frac{1}{1+\omega(1-p_{1})}\). Thus, if \(\delta>x_{1}\), then \(\big{(}R_{\text{aut}}^{M}R_{\text{aut}}^{L}\,,D_{\text{reject}}\big{)}\) is a pure-strategy BNE. Also, if \(\delta>\frac{1}{1+\omega}\), then \(U_{2}\Big{(}R_{\text{cyb}}^{M}R_{\text{aut}}^{L}\,,D_{\text{accept}}\Big{)}<U _{2}\Big{(}R_{\text{cyb}}^{M}R_{\text{aut}}^{L}\,,D_{\text{reject}}\Big{)}\). Let \(y_{1}=\frac{1}{1+\omega}\). If \(\delta>x_{1}\) and \(\delta>y_{1}\), then \(\big{(}R_{\text{aut}}^{M}R_{\text{aut}}^{L}\,,D_{\text{reject}}\big{)}\) is the unique pure-strategy BNE since \(y_{1}<x_{1}\). This implies that if \(\delta>x_{1}\), then \(\big{(}R_{\text{aut}}^{M}R_{\text{aut}}^{L}\,,D_{\text{reject}}\big{)}\) is the unique pure-strategy BNE. Furthermore, if \(\delta<y_{1}\), then \(\Big{(}R_{\text{cyb}}^{M}R_{\text{aut}}^{L}\,,D_{\text{accept}}\Big{)}\) is a pure-strategy BNE. In addition, if \(\delta<y_{1}\) and \(\delta<x_{1}\), then \(\big{(}R_{\text{cyb}}^{M}R_{\text{aut}}^{L}\,,D_{\text{accept}}\big{)}\) is the unique pure-strategy BNE. Since \(y_{1}<x_{1}\), \(\big{(}R_{\text{cyb}}^{M}R_{\text{aut}}^{L}\,,D_{\text{accept}}\big{)}\) is the unique pure-strategy BNE if \(\delta<y_{1}\). If \(\delta<x_{1}\) and \(\delta>y_{1}\), then there exists a mixed-strategy BNE. Let \(\lambda\) denote the probability that an insider chooses the strategy \(R_{\text{aut}}^{M}R_{\text{aut}}^{L}\) and \(\eta\) denote the probability that the defender chooses \(D_{\text{accept}}\). Thus, for the expected values with respect to the pure strategies, we must have \(\mathbb{E}\big{(}R_{\text{aut}}^{M}R_{\text{aut}}^{L}\,\big{)}=\mathbb{E} \Big{(}R_{\text{cyb}}^{M}R_{\text{aut}}^{L}\,\big{)}\). Simplifying this, we obtain  
\[\eta^{*}=\frac{C_{\text{cyb}}}{p_{1}\beta} \tag{1}\]Also, we must have \(\mathbb{E}\big{(}D_{\text{accept}}\ \big{)}=\mathbb{E}\big{(}D_{\text{ reject}}\ \big{)}\). On simplifying this, we obtain  
\[\lambda^{*}=\frac{(1+\omega)-\frac{1}{\delta}}{p_{1}\omega} \tag{10}\]  
Summarizing the results above, we obtain Equation (1).  
(ii) If \(C_{cyb}>\beta p_{1}\), then we have \(U_{1}\big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{accept}}\ \big{)}>U_{1} \big{(}R_{cyb}^{M}\ R_{\text{aut}}^{L}\,D_{\text{accept}}\ \big{)}\) and \(U_{1}\big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{ reject}}\ \big{)}>U_{1} \big{(}R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{ reject}}\ \big{)}\). This shows that the strategy \(R_{\text{cyb}}^{M}\ R_{\text{aut}}^{L}\) is dominated by the strategy \(R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\.\) Thus, if \(\delta>x_{1}\), \(\big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{ reject}}\ \big{)}\) is the unique pure-strategy BNE, whereas, if \(\delta<x_{1}\), then \(\big{(}R_{\text{aut}}^{M}\ R_{\text{aut}}^{L}\,D_{\text{accept}}\ \big{)}\) is the unique pure-strategy BNE. Summarizing the results above, we obtain Equation (2). \(\square\)