## Appendix B Optimal Decision in Scenario RC  
First, we substitute \(m_{r}^{C}=p_{r}^{C}-w_{r}^{C}\) in Equation (1)-(3) to obtain optimal decision in the RS game. To obtain the manufacturer response first, one needs to solve \(\frac{d\pi_{gr}^{C}}{d\rho_{r}^{C}}=0\), \(\frac{d\pi_{gr}^{C}}{d\rho_{r}^{C}}=0\), and \(\frac{d\pi_{gr}^{C}}{d\rho_{r}^{C}}=0\) simultaneously. Therefore, the manufacturer's response is obtained as follows:  
\[w_{r}^{C}=\frac{(\kappa(2\lambda_{2}+Z\lambda_{1})-X^{2}\lambda_{2})(a-m_{r}^ {C}+\rho_{r}^{C})+c_{m}\kappa(2\lambda_{2}-\beta(\beta-\lambda_{1}))}{\kappa( 4\lambda_{2}-Z^{2})-X^{2}\lambda_{2}}\]  
\[\tau=\frac{(a-c_{m}-m_{r}^{C}+\rho_{r}^{C})X\lambda_{2}}{\kappa(4\lambda_{2}- Z^{2})-X^{2}\lambda_{2}}\text{and}\theta=\frac{(a-c_{m}-m_{r}^{C}+\rho_{r}^{C}) \kappa Z}{\kappa(4\lambda_{2}-(\beta-\lambda_{1})^{2})-X^{2}\lambda_{2}}\]  
Because, the profit function for the manufacturer is a function of three variables, therefore, we compute corresponding Hessian matrix(\(H_{r}^{C}\)) for the manufacturer profit function is as follows:  
\[H_{r}^{C}=\left[\begin{array}{cc}\frac{\partial^{2}\pi_{gr}^{C}}{\partial \pi_{gr}^{C}}&\frac{\partial^{2}\pi_{gr}^{C}}{\partial\pi_{gr}^{C}}&\frac{ \partial^{2}\pi_{gr}^{C}}{\partial\pi_{gr}^{C}}\\ \frac{\partial^{2}\pi_{gr}^{C}}{\partial\pi_{gr}^{C}}&\frac{\partial^{2}\pi_{ gr}^{C}}{\partial\pi_{gr}^{C}}&\frac{\partial^{2}\pi_{gr}^{C}}{\partial\pi_{gr}^{C}} \end{array}\right]=\left|\begin{array}{cc}-2&-X&\beta+\lambda_{1}\\ -X&-2\kappa&\beta X\\ \beta+\lambda_{1}&\beta X&-2(\lambda_{2}+\beta\lambda_{1})\end{array}\right|\]  
The principal minors of above Hessian matrix are \(H_{r1}^{C}=-2<0\); \(H_{r2}^{T}=4\kappa-X^{2}>0\); and \(H_{r3}^{T}=-2(\kappa(4\lambda_{2}+Z^{2})-X^{2}\lambda_{2})\), respectively. Consequently, the profit function will be concave if \(4\kappa>X^{2}\) and \(\kappa(4\lambda_{2}+Z^{2})>X^{2}\lambda_{2}\).  
Substituting optimal response for the manufacturer in Equation (1), profit function for the retailer is obtained as \(\pi_{rr}^{C}(m_{r}^{C})=\frac{2m_{r}^{C}\kappa\lambda_{2}(a-c_{m}-m_{r}^{C}+ \rho_{r}^{C})}{\kappa(4\lambda_{2}-Z^{2})-X^{2}\lambda_{2}}\). Therefore, the optimal response for the retailer is obtained by solving \(\frac{d\pi_{gr}^{C}}{d\rho_{r}^{C}}=0\). After simplification, \(m_{r}^{C}=\frac{a-c_{m}+\rho_{r}^{C}}{2}\). The profit function of the retailer is also concave because \(\frac{d^{2}\pi_{gr}^{C}}{d\rho_{r}^{C}}=\frac{-4\kappa\lambda_{2}}{\kappa(4 \lambda_{2}-Z^{2})-X^{2}\lambda_{2}}\). Substituting optimal responses in Equation (3), the simplified value of the SW in Scenario RC is obtained as follows:  
\[\pi_{gr}^{C}(\rho_{r}^{C})=\frac{\kappa\lambda_{2}(a-c_{m}+\rho_{r}^{C})((a-c_{ m})(14\kappa\lambda_{2}-3\kappa Z^{2}-3X^{2}\lambda_{2})+\kappa Z^{2}\rho_{r}^{C}-(2 \kappa-X^{2})\lambda_{2}\rho_{r}^{C})}{4(\kappa(4\lambda_{2}-Z^{2})-X^{2} \lambda_{2})^{2}}\]  
Therefore, the optimal subsidy rate will be obtained by solving \(\frac{d\pi_{gr}^{C}}{d\rho_{r}^{C}}=0\). On simplification, we obtain the value of \(\rho_{r}^{C}\) as presented in Proposition (2). The SW under the RS game is concave because, \(\frac{d^{2}\pi_{gr}^{C}}{d\rho_{r}^{C}}=-\frac{\kappa\lambda_{2}\Delta_{1}}{2( \kappa(4\lambda_{2}-(\beta-\lambda_{1})^{2})-X^{2}\lambda_{2})^{2}}<0\) where \(\Delta_{1}=\kappa(2\lambda_{2}-(\beta-\lambda_{1})^{2})-X^{2}\lambda_{2}\). By using back substitution, we obtain optimal decision as presented in Lemma 2.