Appendix 1
The expected profit of the supply chain is given by
$$\begin{aligned} E\big [{\widetilde{\pi }}_c\big ]=\, & {} E\big [\big (s_1-{\tilde{c}}_1\big )\big (\tilde{a_1}-{\tilde{\beta }} s_1+\tau {\tilde{\beta }} s_2-\theta _1 {\tilde{\beta }} s_3\big )+\big (s_2-{\tilde{c}}_2\big )\big (\tilde{a_2}+\tau {\tilde{\beta }} s_1-{\tilde{\beta }} s_2-\theta _2 {\tilde{\beta }} s_3\big )\nonumber \\&+\big (s_3-{\tilde{c}}_3\big )\big (\tilde{a_3}-\theta _1 {\tilde{\beta }} s_1-\theta _2 {\tilde{\beta }} s_2-{\tilde{\beta }} s_3\big )\big ]\nonumber \\=\, & {} \frac{1}{2}\int _{0}^{1}\big \{\big [\big (s_1-{\tilde{c}}_1\big )\big (\tilde{a_1}-{\tilde{\beta }} s_1+\tau {\tilde{\beta }} s_2-\theta _1 {\tilde{\beta }} s_3\big )\big ]_{\alpha }^{R}+\big [\big (s_1-\tilde{c_1}\big )\big (\tilde{a_1}-{\tilde{\beta }} s_1+\tau {\tilde{\beta }} s_2-\theta _1 {\tilde{\beta }} s_3\big )\big ]_{\alpha }^{L}\nonumber \\&+\big [\big (s_2-{\tilde{c}}_2\big )\big (\tilde{a_2}+\tau {\tilde{\beta }} s_1-{\tilde{\beta }} s_2-\theta _2 {\tilde{\beta }} s_3\big )\big ]_{\alpha }^{R}+\big [\big (s_2-{\tilde{c}}_2\big )\big (\tilde{a_2}+\tau {\tilde{\beta }} s_1-{\tilde{\beta }} s_2-\theta _2 {\tilde{\beta }} s_3\big )\big ]_{\alpha }^{L}\nonumber \\&+\big [\big (s_3-{\tilde{c}}_3\big )\big (\tilde{a_3}-\theta _1 {\tilde{\beta }} s_1-\theta _2 {\tilde{\beta }} s_2-{\tilde{\beta }} s_3\big )\big ]_{\alpha }^{R}+\big [\big (s_3-{\tilde{c}}_3\big )\big (\tilde{a_3}-\theta _1 {\tilde{\beta }} s_1-\theta _2 {\tilde{\beta }} s_2-{\tilde{\beta }} s_3\big )\big ]_{\alpha }^{L}\big \}d\alpha \nonumber \\=\, & {} \frac{1}{2}\int _{0}^{1}\big \{\big (s_1-{\tilde{c}}_{1\alpha }^{L}\big )\big ({\tilde{a}}_{1\alpha }^{R}-{\tilde{\beta }}_{\alpha }^{L} s_1+\tau {\tilde{\beta }}_{\alpha }^{R} s_2-\theta _1 {\tilde{\beta }}_{\alpha }^{L} s_3\big )+\big [\big (s_1-{\tilde{c}}_{1\alpha }^{R}\big )\big ({\tilde{a}}_{1\alpha }^{L}-{\tilde{\beta }}_{\alpha }^{R} s_1+\tau {\tilde{\beta }}_{\alpha }^{L} s_2\nonumber \\&-\theta _1 {\tilde{\beta }}_{\alpha }^{R} s_3\big )\big ]+\big (s_2-{\tilde{c}}_{2\alpha }^{L}\big )\big ({\tilde{a}}_{2\alpha }^{R}+\tau {\tilde{\beta }}_{\alpha }^{R} s_1-{\tilde{\beta }}_{\alpha }^{L} s_2-\theta _2 {\tilde{\beta }}_{\alpha }^{L} s_3\big )+\big (s_2-{\tilde{c}}_{2\alpha }^{R}\big )\big ({\tilde{a}}_{2\alpha }^{L}+\tau {\tilde{\beta }}_{\alpha }^{L} s_1\nonumber \\&-{\tilde{\beta }}_{\alpha }^{R} s_2-\theta _2 {\tilde{\beta }}_{\alpha }^{R} s_3\big )+\big (s_3-{\tilde{c}}_{3\alpha }^{L}\big )\big ({\tilde{a}}_{3\alpha }^{R}-\theta _1 {\tilde{\beta }}_{\alpha }^{L} s_1-\theta _2 {\tilde{\beta }}_{\alpha }^{L} s_2-{\tilde{\beta }}_{\alpha }^{L} s_3\big )\nonumber \\&+\big (s_3-{\tilde{c}}_{3\alpha }^{R}\big )\big ({\tilde{a}}_{3\alpha }^{L}-\theta _1 {\tilde{\beta }}_{\alpha }^{R} s_1-\theta _2 {\tilde{\beta }}_{\alpha }^{R} s_2-{\tilde{\beta }}_{\alpha }^{R} s_3\big )\big \}d\alpha \nonumber \\=\, & {} \frac{1}{2}\int _{0}^{1}\big \{-\big ({\tilde{\beta }}_{\alpha }^{L}+{\tilde{\beta }}_{\alpha }^{R}\big )s_1^2-\big ({\tilde{\beta }}_{\alpha }^{L}+{\tilde{\beta }}_{\alpha }^{R}\big )s_2^2-\big ({\tilde{\beta }}_{\alpha }^{L}+{\tilde{\beta }}_{\alpha }^{R}\big )s_3^2+2\tau \big ({\tilde{\beta }}_{\alpha }^{L}+{\tilde{\beta }}_{\alpha }^{R}\big )s_1s_2\nonumber \\&-2\theta _1 \big ({\tilde{\beta }}_{\alpha }^{L}+{\tilde{\beta }}_{\alpha }^{R}\big )s_1s_3-2\theta _2\big ({\tilde{\beta }}_{\alpha }^{L}+{\tilde{\beta }}_{\alpha }^{R}\big )s_2s_3+ \big [\big ({\tilde{a}}_{1\alpha }^{R}+{\tilde{a}}_{1\alpha }^{L}\big )+\big ({\tilde{c}}_{1\alpha }^{L}{\tilde{\beta }}_{\alpha }^{L}+{\tilde{c}}_{1\alpha }^{R}{\tilde{\beta }}_{\alpha }^{R}\big )\nonumber \\&-\tau \big ({\tilde{c}}_{2\alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{2\alpha }^{R}{\tilde{\beta }}_{\alpha }^{L}\big )+ \theta _1 \big ({\tilde{c}}_{3\alpha }^{L}{\tilde{\beta }}_{\alpha }^{L}+{\tilde{c}}_{3\alpha }^{R}{\tilde{\beta }}_{\alpha }^{R}\big )\big ]s_1 +\big [\big ({\tilde{a}}_{2\alpha }^{R}+{\tilde{a}}_{2\alpha }^{L}\big )-\tau \big ({\tilde{c}}_{1\alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{1\alpha }^{R}{\tilde{\beta }}_{\alpha }^{L}\big )\nonumber \\&+\big ({\tilde{c}}_{2\alpha }^{L}{\tilde{\beta }}_{\alpha }^{L}+{\tilde{c}}_{2\alpha }^{R}{\tilde{\beta }}_{\alpha }^{R}\big )+ \theta _2 \big ({\tilde{c}}_{3\alpha }^{L}{\tilde{\beta }}_{\alpha }^{L}+{\tilde{c}}_{3\alpha }^{R}{\tilde{\beta }}_{\alpha }^{R}\big )\big ]s_2+\big [\big ({\tilde{a}}_{3\alpha }^{R}+{\tilde{a}}_{3\alpha }^{L}\big )+\theta _1 \big ({\tilde{c}}_{1\alpha }^{L}{\tilde{\beta }}_{\alpha }^{L}+{\tilde{c}}_{1\alpha }^{R}{\tilde{\beta }}_{\alpha }^{R}\big )\nonumber \\&+\theta _2 \big ({\tilde{c}}_{2\alpha }^{L}{\tilde{\beta }}_{\alpha }^{L}+{\tilde{c}}_{2\alpha }^{R}{\tilde{\beta }}_{\alpha }^{R}\big )+\big ({\tilde{c}}_{3\alpha }^{L}{\tilde{\beta }}_{\alpha }^{L}+{\tilde{c}}_{3\alpha }^{R}{\tilde{\beta }}_{\alpha }^{R}\big )\big ]s_3-\big [\big ({\tilde{c}}_{1\alpha }^{L}{\tilde{a}}_{1\alpha }^{R}+{\tilde{c}}_{1\alpha }^{R}{\tilde{a}}_{1\alpha }^{L}\big )\nonumber \\&+ \big ({\tilde{c}}_{2\alpha }^{L}{\tilde{a}}_{2\alpha }^{R}+{\tilde{c}}_{2\alpha }^{R}{\tilde{a}}_{2\alpha }^{L}\big )+\big ({\tilde{c}}_{3\alpha }^{L}{\tilde{a}}_{3\alpha }^{R}+{\tilde{c}}_{3\alpha }^{R}{\tilde{a}}_{3\alpha }^{L}\big )\big ]\big \}d\alpha \nonumber \\=\, & {} -E\big [{\tilde{\beta }}\big ]s_1^2-E\big [{\tilde{\beta }}\big ]s_2^2-E\big [{\tilde{\beta }}\big ]s_3^2+2\tau E\big [{\tilde{\beta }}\big ]s_1s_2-2\theta _1 E\big [{\tilde{\beta }}\big ]s_1s_3-2\theta _2 E\big [{\tilde{\beta }}\big ]s_2s_3\nonumber \\&+\big (E\big [{\tilde{a}}_{1}\big ]+E\big [{\tilde{c}}_{1}{\tilde{\beta }}\big ]+\theta _1 E\big [{\tilde{c}}_{3}{\tilde{\beta }}\big ]\big )s_1+\big (E\big [{\tilde{a}}_{2}\big ]+E\big [{\tilde{c}}_{2}{\tilde{\beta }}\big ]+\theta _2 E\big [{\tilde{c}}_{3}{\tilde{\beta }}\big ]\big )s_2+\big (E\big [{\tilde{a}}_{3}\big ]\nonumber \\&+\theta _1 E\big [{\tilde{c}}_{1}{\tilde{\beta }}\big ]+\theta _2 E\big [{\tilde{c}}_{2}{\tilde{\beta }}\big ]+E\big [{\tilde{c}}_{3}{\tilde{\beta }}\big ]\big )s_3-\frac{\tau }{2}\sum _{i=1}^{2}\left[ \int _0^1\left( {\tilde{c}}_{i \alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{i \alpha }^{R}{\tilde{\beta }}_{\alpha }^{L}\right) d\alpha \right] s_{(3-i)}\nonumber \\&-\frac{1}{2}\sum _{j=1}^3\int _0^1\left( {\tilde{c}}_{j \alpha }^{L}{\tilde{a}}_{j\alpha }^{R}+{\tilde{c}}_{j \alpha }^{R}{\tilde{a}}_{j \alpha }^{L}\right) d\alpha \end{aligned}$$
(20)
The first and second order partial diff. of Eq. (20) w. r. to \(s_1\), \(s_2\) and \(s_3\) are as follows
$$\begin{aligned} \frac{\partial E\left[ {\widetilde{\pi }}_c\right] }{\partial s_1}=\, & {} -2E\left[ {\tilde{\beta }}\right] s_1+2\tau E\left[ {\tilde{\beta }}\right] s_2-2\theta _1E\left[ {\tilde{\beta }}\right] s_3+E\left[ {\tilde{a}}_{1}\right] +E\left[ {\tilde{c}}_{1}{\tilde{\beta }}\right] +\theta _1 E\left[ {\tilde{c}}_{3}{\tilde{\beta }}\right] -\frac{\tau }{2}\int _0^1\left( {\tilde{c}}_{2 \alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{2 \alpha }^{R}{\tilde{\beta }}_{\alpha }^{L}\right) d\alpha ,\\ \frac{\partial E\left[ {\widetilde{\pi }}_c\right] }{\partial s_2}=\, & {} 2\tau E\left[ {\tilde{\beta }}\right] s_1-2E\left[ {\tilde{\beta }}\right] s_2-2\theta _2 E\left[ {\tilde{\beta }}\right] s_3+E\left[ {\tilde{a}}_{2}\right] +E\left[ {\tilde{c}}_{2}{\tilde{\beta }}\right] +\theta _2 E\left[ {\tilde{c}}_{3}{\tilde{\beta }}\right] -\frac{\tau }{2}\int _0^1\left( {\tilde{c}}_{1 \alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{1 \alpha }^{R}{\tilde{\beta }}_{\alpha }^{L}\right) d\alpha ,\\ \frac{\partial E\left[ {\widetilde{\pi }}_c\right] }{\partial s_3}= \,& {} -2\theta _1E\left[ {\tilde{\beta }}\right] s_1-2\theta _2E\left[ {\tilde{\beta }}\right] s_2-2E\left[ {\tilde{\beta }}\right] s_3+E\left[ {\tilde{a}}_{3}\right] +\theta _1 E\left[ {\tilde{c}}_{1}{\tilde{\beta }}\right] +\theta _2 E\left[ {\tilde{c}}_{2}{\tilde{\beta }}\right] +E\left[ {\tilde{c}}_{3}{\tilde{\beta }}\right] ,\\ \frac{\partial ^2 E\left[ {\widetilde{\pi }}_c\right] }{\partial s^2_1}=\, & {} -2E\left[ {\tilde{\beta }}\right] ,\frac{\partial ^2 E\left[ {\widetilde{\pi }}_c\right] }{\partial s_2 \partial s_1}=2\tau E\left[ {\tilde{\beta }}\right] ,\frac{\partial ^2 E\left[ {\widetilde{\pi }}_c\right] }{\partial s_3 \partial s_1}=-2\theta _1E\left[ {\tilde{\beta }}\right] ,\frac{\partial ^2 E\left[ {\widetilde{\pi }}_c\right] }{\partial s_1 \partial s_2}=2\tau E\left[ {\tilde{\beta }}\right] ,\frac{\partial ^2 E\left[ {\widetilde{\pi }}_c\right] }{\partial s^2_2}=-2E\left[ {\tilde{\beta }}\right] ,\\ \frac{\partial ^2 E\left[ {\widetilde{\pi }}_c\right] }{\partial s_3 \partial s_2}=\, & {} -2\theta _2E\left[ {\tilde{\beta }}\right] ,\frac{\partial ^2 E\left[ {\widetilde{\pi }}_c\right] }{\partial s_1 \partial s_3}=-2\theta _1E\left[ {\tilde{\beta }}\right] ,\frac{\partial ^2 E\left[ {\widetilde{\pi }}_c\right] }{\partial s_2 \partial s_3}=-2\theta _2E\left[ {\tilde{\beta }}\right] \;{\mathrm{and}}\;\frac{\partial ^2 E\left[ {\widetilde{\pi }}_c\right] }{\partial s^2_3}=-2E\left[ {\tilde{\beta }}\right] . \end{aligned}$$
Determinant value of the Hessian matrix |H| is given by
$$\begin{aligned} \left| \begin{array}{lll} -2E[{\tilde{\beta }}] &{} 2\tau E[{\tilde{\beta }}] &{} -2\theta _1 E[{\tilde{\beta }}]\\ 2\tau E[{\tilde{\beta }}] &{} -2E[{\tilde{\beta }}] &{} -2\theta _2 E[{\tilde{\beta }}] \\ -2\theta _1 E[{\tilde{\beta }}] &{} -2\theta _2 E[{\tilde{\beta }}] &{} -2E[{\tilde{\beta
}}]\end{array}\right| =8\{E[{\tilde{\beta }}]\}^3(\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1). \end{aligned}$$
Therefore, the profit expression given in Eq. (20) is concave if \(\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2<1\) and \(\tau ^2<1\).
At the extreme point, we have
$$\begin{aligned} \begin{aligned} -2E[{\tilde{\beta }}]s^*_1+2\tau E[{\tilde{\beta }}] s^*_2-2\theta _1 E[{\tilde{\beta }}] s^*_3&=-K_1,\\ \,\,\,\,2\tau E[{\tilde{\beta }}] s^*_1-2E[{\tilde{\beta }}]s^*_2-2\theta _2 E[{\tilde{\beta }}] s^*_3&=-K_2,\\ -2\theta _1 E[{\tilde{\beta }}] s^*_1-2\theta _2 E[{\tilde{\beta }}] s^*_2-2E[{\tilde{\beta }}]s^*_3&=-K_3, \end{aligned} \end{aligned}$$
(21)
where \(K_1=E[{\tilde{a}}_{1}]+E[{\tilde{c}}_{1}{\tilde{\beta }}]+\theta _1 E[{\tilde{c}}_{3}{\tilde{\beta }}]-\frac{\tau }{2}\int _0^1({\tilde{c}}_{2 \alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{2 \alpha }^{R}{\tilde{\beta }}_{\alpha }^{L})d\alpha\), \(K_2=E[{\tilde{a}}_{2}]+E[{\tilde{c}}_{2}{\tilde{\beta }}]+\theta _2 E[{\tilde{c}}_{3}{\tilde{\beta }}]-\frac{\tau }{2}\int _0^1({\tilde{c}}_{1 \alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{1 \alpha }^{R}{\tilde{\beta }}_{\alpha }^{L})d\alpha\) and \(K_3=E[{\tilde{a}}_{3}]+\theta _1 E[{\tilde{c}}_{1}{\tilde{\beta }}]+\theta _2 E[{\tilde{c}}_{2}{\tilde{\beta }}]+E[{\tilde{c}}_{3}{\tilde{\beta }}].\)
Solving the system of linear Eqs. given in (21), we have
$$\begin{aligned} s^*_1=-\frac{(1-\theta _2^2)K_1+(\tau +\theta _1\theta _2)K_2-(\theta _1+\tau \theta _2)K_3}{2E[{\tilde{\beta }}](\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1)} \end{aligned}$$
(22)
$$\begin{aligned} s^*_2=-\frac{(\tau +\theta _1\theta _2)K_1+(1-\theta _1^2)K_2-(\theta _2+\tau \theta _1)K_3}{2E[{\tilde{\beta }}](\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1)} \end{aligned}$$
(23)
$$\begin{aligned} s^*_3=\frac{(\theta _1+\tau \theta _2)K_1+(\theta _2+\tau \theta _1)K_2-(1-\tau ^2)K_3}{2E[{\tilde{\beta }}](\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1)} \end{aligned}$$
(24)
From the constraint \({{\textit{Pos}}}(\{\tilde{a_1}\ge {\tilde{\beta }} s^*_1-\tau {\tilde{\beta }} s^*_2+\theta _1{\tilde{\beta }} s^*_3\})>\eta _1\) and using Definitions 3 and 4, we have
$$\begin{aligned} {\tilde{a}}^{R}_{1\eta _1}>{\tilde{\beta }}^{L}_{\eta _1} s^*_1-\tau {\tilde{\beta }}^{R}_{\eta _1} s^*_2+\theta _1{\tilde{\beta }}^{L}_{\eta _1} s^*_3. \end{aligned}$$
Similarly, from the constraints \({{\textit{Pos}}}(\{\tilde{a_2}\ge -\tau {\tilde{\beta }} s^*_1 +{\tilde{\beta }} s^*_2+\theta _2{\tilde{\beta }} s^*_3\})>\eta _1\), \({{\textit{Pos}}}(\{\tilde{a_3}\ge \theta _1{\tilde{\beta }} s^*_1+\theta _2{\tilde{\beta }} s^*_2+{\tilde{\beta }} s^*_3\})>\eta _1\) and \({{\textit{Pos}}}(\{\tilde{c_i} \le s^*_i\})>\eta _2,\,i=1,2,3\), we have
$$\begin{aligned} {\tilde{a}}^{R}_{2\eta _1}>-\tau {\tilde{\beta }}^{R}_{\eta _1} s^*_1+{\tilde{\beta }}^{L}_{\eta _1} s^*_2 +\theta _2{\tilde{\beta }}^{L}_{\eta _1} s^*_3,\,\,{\tilde{a}}^{R}_{3\eta _1}>\theta _1{\tilde{\beta }}^{L}_{\eta _1} s^*_1+\theta _2{\tilde{\beta }}^{L}_{\eta _1} s^*_2+{\tilde{\beta }}^{L}_{\eta _1} s^*_3 \text{ and } {\tilde{c}}^{L}_{i\eta _2}<s^*_i,\,i=1,2,3. \end{aligned}$$
Appendix 2
The expected profit of the retailer in the MS approach is
$$\begin{aligned} E[{\widetilde{\pi }}_r]=\, & {} (s_1-p_1)(E[\tilde{a_1}]-E[{\tilde{\beta }}] s_1+\tau E[{\tilde{\beta }}] s_2-\theta _1 E[{\tilde{\beta }}] s_3)+(s_2-p_2)(E[\tilde{a_2}]+\tau E[{\tilde{\beta }}] s_1\nonumber \\&-E[{\tilde{\beta }}] s_2-\theta _2 E[{\tilde{\beta }}] s_3)+(s_3-p_3)(E[\tilde{a_3}]-\theta _1 E[{\tilde{\beta }}] s_1-\theta _2 E[{\tilde{\beta }}] s_2-E[{\tilde{\beta }}] s_3) \end{aligned}$$
(25)
Proceeding as “Appendix 1”, retailer’s expected profit expression in the MS approach is concave if
$$\begin{aligned} \tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2<1 \text{ and } \tau ^2<1 \end{aligned}$$
and the retailer’s optimal pricing decisions become
$$\begin{aligned} s^*_1=\frac{p_1}{2}-\frac{(1-\theta _2^2)E[{\tilde{a}}_1]+(\tau +\theta _1\theta _2)E[{\tilde{a}}_2]-(\theta _1+\tau \theta _2)E[{\tilde{a}}_3]}{2E[{\tilde{\beta }}](\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1)} \end{aligned}$$
(26)
$$\begin{aligned} s^*_2=\frac{p_2}{2}-\frac{(\tau +\theta _1\theta _2)E[{\tilde{a}}_1]+(1-\theta _1^2)E[{\tilde{a}}_2]-(\theta _2+\tau \theta _1)E[{\tilde{a}}_3]}{2E[{\tilde{\beta }}](\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1)} \end{aligned}$$
(27)
$$\begin{aligned} s^*_3=\frac{p_3}{2}+\frac{(\theta _1+\tau \theta _2)E[{\tilde{a}}_1]+(\theta _2+\tau \theta _1)E[{\tilde{a}}_2]-(1-\tau ^2)E[{\tilde{a}}_3]}{2E[{\tilde{\beta }}](\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1)} \end{aligned}$$
(28)
From the constraints \({{\textit{Pos}}}(\{\tilde{a_1}\ge {\tilde{\beta }} s^*_1-\tau {\tilde{\beta }} s^*_2+\theta _1{\tilde{\beta }} s^*_3\})>\eta _1\), \({{\textit{Pos}}}(\{\tilde{a_2}\ge -\tau {\tilde{\beta }} s^*_1 +{\tilde{\beta }} s^*_2+\theta _2{\tilde{\beta }} s^*_3\})>\eta _1\), \({{\textit{Pos}}}(\{\tilde{a_3}\ge \theta _1{\tilde{\beta }} s^*_1+\theta _2{\tilde{\beta }} s^*_2+{\tilde{\beta }} s^*_3\})>\eta _1\) and \(s^*_i > p_i\) (i=1,2,3) as similar as “Appendix 1”, we have
$$\begin{aligned}&{\tilde{a}}^{R}_{1\eta _1}>{\tilde{\beta }}^{L}_{\eta _1} s^*_1-\tau {\tilde{\beta }}^{R}_{\eta _1} s^*_2+\theta _1{\tilde{\beta }}^{L}_{\eta _1} s^*_3,\,\, {\tilde{a}}^{R}_{2\eta _1}>-\tau {\tilde{\beta }}^{R}_{\eta _1} s^*_1+{\tilde{\beta }}^{L}_{\eta _1} s^*_2 +\theta _2{\tilde{\beta }}^{L}_{\eta _1} s^*_3,\,\,{\tilde{a}}^{R}_{3\eta _1}>\theta _1{\tilde{\beta }}^{L}_{\eta _1} s^*_1+\theta _2{\tilde{\beta }}^{L}_{\eta _1} s^*_2+{\tilde{\beta }}^{L}_{\eta _1} s^*_3,\\&E[{\tilde{\beta }}](\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1)p_1+(1-\theta _2^2)E[{\tilde{a}}_1]+(\tau +\theta _1\theta _2)E[{\tilde{a}}_2]>(\theta _1+\tau \theta _2)E[{\tilde{a}}_3],\\&E[{\tilde{\beta }}](\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1)p_2+(\tau +\theta _1\theta _2)E[{\tilde{a}}_1]+(1-\theta _1^2)E[{\tilde{a}}_2]>(\theta _2+\tau \theta _1)E[{\tilde{a}}_3],\\ \text{ and }&E[{\tilde{\beta }}](\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1)p_3+(1-\tau ^2)E[{\tilde{a}}_3]>(\theta _1+\tau \theta _2)E[{\tilde{a}}_1]+(\theta _2+\tau \theta _1)E[{\tilde{a}}_2]. \end{aligned}$$
By substituting the Eqs. (26), (27) and (28) into the Eqs. (4), (5) and (6), the manufacturers’ expected profits can be expressed as
$$\begin{aligned} E\left[ {\widetilde{\pi }}_{M_1}\right]=\, & {} -\frac{1}{2}E\left[ {\tilde{\beta }}\right] p_1^2+\frac{\tau }{2} E\left[ {\tilde{\beta }}\right] p_1p_2-\frac{\theta _1}{2} E\left[ {\tilde{\beta }}\right] p_1p_3+ \frac{1}{2}\left\{ E\left[ {\tilde{a}}_{1}\right] +E\left[ {\tilde{c}}_{1}{\tilde{\beta }}\right] \right\} p_1\nonumber \\&-\frac{\tau }{4}\left[ \int _0^1\left( {\tilde{c}}_{1 \alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{1 \alpha }^{R}{\tilde{\beta }}_{\alpha }^{L}\right) d\alpha \right] p_2+\frac{\theta _1}{2} E\left[ {\tilde{c}}_{1}{\tilde{\beta }}\right] p_3+\frac{E\left[ {\tilde{c}}_{1}{\tilde{\beta }}\right] E\left[ {\tilde{a}}_{1}\right] }{2E\left[ {\tilde{\beta }}\right] }-\frac{1}{2}\int _0^1\left( {\tilde{c}}_{1 \alpha }^{L}{\tilde{a}}_{1\alpha }^{R}+{\tilde{c}}_{1 \alpha }^{R}{\tilde{a}}_{1 \alpha }^{L}\right) d\alpha \end{aligned}$$
(29)
$$\begin{aligned} E\left[ {\widetilde{\pi }}_{M_2}\right]& {} =\, \frac{\tau }{2} E\left[ {\tilde{\beta }}\right] p_1p_2-\frac{1}{2}E\left[ {\tilde{\beta }}\right] p_2^2-\frac{\theta _2}{2} E\left[ {\tilde{\beta }}\right] p_2p_3-\frac{\tau }{4}\left[ \int _0^1\left( {\tilde{c}}_{2 \alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{2 \alpha }^{R}{\tilde{\beta }}_{\alpha }^{L}\right) d\alpha \right] p_1\nonumber \\&\quad +\frac{1}{2}\left\{ E\left[ {\tilde{a}}_{2}\right] +E\left[ {\tilde{c}}_{2}{\tilde{\beta }}\right] \right\} p_2+\frac{\theta _2}{2} E\left[ {\tilde{c}}_{2}{\tilde{\beta }}\right] p_3+\frac{E\left[ {\tilde{c}}_{2}{\tilde{\beta }}\right] E\left[ {\tilde{a}}_{2}\right] }{2E\left[ {\tilde{\beta }}\right] }-\frac{1}{2}\int _0^1\left( {\tilde{c}}_{2 \alpha }^{L}{\tilde{a}}_{2\alpha }^{R}+{\tilde{c}}_{2 \alpha }^{R}{\tilde{a}}_{2 \alpha }^{L}\right) d\alpha \end{aligned}$$
(30)
$$\begin{aligned} E\left[ {\widetilde{\pi }}_{M_3}\right]=\, & {} -\frac{1}{2}\theta _1 E\left[ {\tilde{\beta }}\right] p_1p_3-\frac{1}{2}\theta _2 E\left[ {\tilde{\beta }}\right] p_2p_3- \frac{1}{2}E\left[ {\tilde{\beta }}\right] p_3^2+\frac{\theta _1}{2} E\left[ {\tilde{c}}_{3}{\tilde{\beta }}\right] p_1+\frac{\theta _2}{2} E\left[ {\tilde{c}}_{3}{\tilde{\beta }}\right] p_2\nonumber \\&+\frac{1}{2}\left\{ E\left[ {\tilde{a}}_{3}\right] +E\left[ {\tilde{c}}_{3}{\tilde{\beta }}\right] \right\} p_3+\frac{E\left[ {\tilde{c}}_{3}{\tilde{\beta }}\right] E\left[ {\tilde{a}}_{3}\right] }{2E\left[ {\tilde{\beta }}\right] }-\frac{1}{2}\int _0^1\left( {\tilde{c}}_{3 \alpha }^{L}{\tilde{a}}_{3\alpha }^{R}+{\tilde{c}}_{3 \alpha }^{R}{\tilde{a}}_{3\alpha }^{L}\right) d\alpha \end{aligned}$$
(31)
The first and second order partial diff. of Eq. (29) w. r. to \(p_1\) are as follows
$$\begin{aligned} \frac{\partial E[{\widetilde{\pi }}_{M_1}]}{\partial p_1}=-E[{\tilde{\beta }}] p_1+\frac{\tau }{2}E[{\tilde{\beta }}]p_2-\frac{\theta _1}{2}E[{\tilde{\beta }}]p_3+\frac{1}{2}(E[{\tilde{a}}_1]+E[{\tilde{\beta }}{\tilde{c}}_1]),\,\,and\,\,\frac{ \partial ^2 E[{\widetilde{\pi }}_{M_1}]}{\partial p^2_1}=-E[{\tilde{\beta }}]<0 \end{aligned}$$
Therefore, the manufacturer’s (\(M_1\)) profit expression in MS approach is concave as \(E[{\tilde{\beta }}]>0\) and at the extreme point, we have
$$\begin{aligned} -E[{\tilde{\beta }}] p^*_1+\frac{\tau }{2}E[{\tilde{\beta }}]p^*_2-\frac{\theta _1}{2}E[{\tilde{\beta }}]p^*_3=-\frac{1}{2}(E[{\tilde{a}}_1]+E[{\tilde{\beta }}{\tilde{c}}_1]). \end{aligned}$$
(32)
Similarly, it can be shown that manufacturers’ (\(M_2\) and \(M_3\)) profit expressions for MS approach are also concave and at the extreme point they become
$$\begin{aligned} \frac{\tau }{2}E[{\tilde{\beta }}]p^*_1-E[{\tilde{\beta }}] p^*_2-\frac{\theta _2}{2}E[{\tilde{\beta }}]p^*_3=\, & {} -\frac{1}{2}(E[{\tilde{a}}_2]+E[{\tilde{\beta }}{\tilde{c}}_2]) \end{aligned}$$
(33)
$$\begin{aligned} -\frac{\theta _1}{2}E[{\tilde{\beta }}] p^*_1-\frac{\theta _2}{2}E[{\tilde{\beta }}]p^*_2-E[{\tilde{\beta }}] p^*_3=\, & {} -\frac{1}{2}(E[{\tilde{a}}_3]+E[{\tilde{\beta }}{\tilde{c}}_3]) \end{aligned}$$
(34)
Solving Eqs. (32), (33) and (34), we have
$$\begin{aligned} p^*_1=\, & {} -\frac{(4-\theta _2^2)(E[{\tilde{a}}_1]+E[{\tilde{\beta }}{\tilde{c}}_1])+(2\tau +\theta _1\theta _2)(E[{\tilde{a}}_2]+E[{\tilde{\beta }}{\tilde{c}}_2])-(2\theta _1+\tau \theta _2)(E[{\tilde{a}}_3]+E[{\tilde{\beta }}{\tilde{c}}_3])}{2E[{\tilde{\beta }}](\tau ^2+\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-4)} \end{aligned}$$
(35)
$$\begin{aligned} p^*_2=\, & {} -\frac{(2\tau +\theta _1\theta _2)(E[{\tilde{a}}_1]+E[{\tilde{\beta }}{\tilde{c}}_1])+(4-\theta _1^2)(E[{\tilde{a}}_2]+E[{\tilde{\beta }}{\tilde{c}}_2])-(2\theta _2+\tau \theta _1)(E[{\tilde{a}}_3]+E[{\tilde{\beta }}{\tilde{c}}_3])}{2E[{\tilde{\beta }}](\tau ^2+\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-4)}\end{aligned}$$
(36)
$$\begin{aligned} p^*_3=\, & {} \frac{(2\theta _1+\tau \theta _2)(E[{\tilde{a}}_1]+E[{\tilde{\beta }}{\tilde{c}}_1])+(2\theta _2+\tau \theta _1)(E[{\tilde{a}}_2]+E[{\tilde{\beta }}{\tilde{c}}_2])-(4-\tau ^2)(E[{\tilde{a}}_3]+E[{\tilde{\beta }}{\tilde{c}}_3])}{2E[{\tilde{\beta }}](\tau ^2+\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-4)} \end{aligned}$$
(37)
From the constraints \({{\textit{Pos}}}(\{\tilde{c_i} \le p^*_i\})>\eta _2,\,i=1,2,3\), we have \({\tilde{c}}^{L}_{i\eta _2}<p^*_i,\,i=1,2,3.\)
Appendix 3
The expected profit of the manufacturer \(M_1\) is
$$\begin{aligned} E\left[ {\widetilde{\pi }}_{M_1}\right]= & {} -E\left[ {\tilde{\beta }}\right] p_1^2+\tau E\left[ {\tilde{\beta }}\right] p_1p_2-\theta _1 E\left[ {\tilde{\beta }}\right] p_1p_3+ \left\{ E\left[ {\tilde{a}}_{1}\right] -\left( m_1-\tau m_2+ \theta _1 m_3\right) E\left[ {\tilde{\beta }}\right] \right. \nonumber \\&\left. +\,E\left[ {\tilde{c}}_{1}{\tilde{\beta }}\right] \right\} p_1-\frac{\tau }{2}\left[ \int _0^1\left( {\tilde{c}}_{1 \alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{1 \alpha }^{R}{\tilde{\beta }}_{\alpha }^{L}\right) d\alpha \right] p_2+\theta _1 E\left[ {\tilde{c}}_{1}{\tilde{\beta }}\right] p_3+E\left[ {\tilde{c}}_{1}{\tilde{\beta }}\right] m_1\nonumber \\&+\theta _1 E\left[ {\tilde{c}}_{1}{\tilde{\beta }}\right] m_3-\frac{\tau m_2}{2}\left[ \int _0^1\left( {\tilde{c}}_{1 \alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{1 \alpha }^{R}{\tilde{\beta }}_{\alpha }^{L}\right) d\alpha \right] -\frac{1}{2}\int _0^1\left( {\tilde{c}}_{1 \alpha }^{L}{\tilde{a}}_{1\alpha }^{R}+{\tilde{c}}_{1 \alpha }^{R}{\tilde{a}}_{1 \alpha }^{L}\right) d\alpha \end{aligned}$$
(38)
The first and second order partial diff. of Eq. (38) w. r. to \(p_1\) are as follows
$$\begin{aligned} \frac{\partial E\left[ {\widetilde{\pi }}_{M_1}\right] }{\partial p_1}=E\left[ {\tilde{a}}_1\right] -E\left[ {\tilde{\beta }}\right] (p_1+m_1)+\tau E\left[ {\tilde{\beta }}\right] (p_2+m_2)-\theta _1E\left[ {\tilde{\beta }}\right] (p_3+m_3)-E\left[ {\tilde{\beta }}\right] p_1+E\left[ {\tilde{\beta }}{\tilde{c}}_1\right] \end{aligned}$$
and \(\frac{ \partial ^2 E[{\widetilde{\pi }}_{M_1}]}{\partial p^2_1}=-2E[{\tilde{\beta }}]<0.\) Therefore, the manufacturer’s (\(M_1\)) expected profit expression in RS approach is concave as \(E[{\tilde{\beta }}]>0\) and at the extreme point, optimum pricing decisions of \(M_1\) is
$$\begin{aligned} p^*_1=\frac{E[{\tilde{a}}_1]+E[{\tilde{\beta }}{\tilde{c}}_1]}{E[{\tilde{\beta }}]}-s_1+\tau s_2-\theta _1 s_3. \end{aligned}$$
(39)
Similarly, we can show that manufacturers’ (\(M_2\) and \(M_3\)) expected profit expressions in RS approach are concave as \(E[{\tilde{\beta }}]>0\) and optimum pricing decisions for \(M_2\) and \(M_3\) are respectively given by
$$\begin{aligned} p^*_2= & {} \frac{E[{\tilde{a}}_2]+E[{\tilde{\beta }}{\tilde{c}}_2]}{E[{\tilde{\beta }}]}+\tau s_1- s_2-\theta _2 s_3 \end{aligned}$$
(40)
$$\begin{aligned} p^*_3= & {} \frac{E[{\tilde{a}}_3]+E[{\tilde{\beta }}{\tilde{c}}_3]}{E[{\tilde{\beta }}]}-\theta _1 s_1-\theta _2 s_2- s_3 \end{aligned}$$
(41)
From the constraints \({{\textit{Pos}}}(\{\tilde{c_i} \le p^*_i\})>\eta _2,\,i=1,2,3\), we have \({\tilde{c}}^{L}_{i\eta _2}<p^*_i,\,i=1,2,3.\)
The retailer’s expected profit after manufacturers’ optimal decisions in RS approach is given by
$$\begin{aligned} E[{\widetilde{\pi }}_r]= & {} (s_1-p^*_1)(a_1-E[{\tilde{\beta }}] s_1+\tau E[{\tilde{\beta }}] s_2-\theta _1 E[{\tilde{\beta }}] s_3)+(s_2-p^*_2)(a_2 +\tau E[{\tilde{\beta }}] s_1-E[{\tilde{\beta }}] s_2\nonumber \\&-\theta _2 E[{\tilde{\beta }}] s_3) +(s_3-p^*_3)(a_3-\theta _1 E[{\tilde{\beta }}] s_1-\theta _2 E[{\tilde{\beta }}] s_2-E[{\tilde{\beta }}] s_3) \end{aligned}$$
(42)
The first and second order partial diff. of Eq. (42) w. r. to \(s_1\), \(s_2\) and \(s_3\) are as follows
$$\begin{aligned} \frac{\partial E[{\widetilde{\pi }}_{r}]}{\partial s_1} & {} = -P s_1+Q s_2-R s_3+B_1,\,\,\,\frac{\partial E[{\widetilde{\pi }}_{r}]}{\partial s_2}=Q s_1-S s_2-T s_3+B_2,\\ \frac{\partial E[{\widetilde{\pi }}_{r}]}{\partial s_3}& {} = -R s_1-T s_2-U s_3+B_3,\,\,\frac{\partial ^2 E[{\widetilde{\pi }}_{r}]}{\partial s^2_1}=-P,\,\,\frac{\partial ^2 E[{\widetilde{\pi }}_{r}]}{\partial s_2 \partial s_1}=Q,\,\,\frac{\partial ^2 E[{\widetilde{\pi }}_{r}]}{\partial s_3 \partial s_1}=-R,\\ \frac{\partial ^2 E[{\widetilde{\pi }}_{r}]}{\partial s_1 \partial s_2}& {} = Q,\,\,\frac{\partial ^2 E[{\widetilde{\pi }}_{r}]}{\partial s^2_2}=-S,\,\,\frac{\partial ^2 E[{\widetilde{\pi }}_{r}]}{\partial s_3 \partial s_2}=-T,\,\,\frac{\partial ^2 E[{\widetilde{\pi }}_{r}]}{\partial s_1 \partial s_3}=-R,\,\,\frac{\partial ^2 E[{\widetilde{\pi }}_{r}]}{\partial s_2 \partial s_3}=-T,\,\,\, \frac{\partial ^2 E[{\widetilde{\pi }}_{r}]}{\partial s^2_3}=-U,\\ \text{ where } P& {} = 2E[{\tilde{\beta }}](2+\tau ^2+\theta _1^2),\,\,Q= 2E[{\tilde{\beta }}](3\tau -\theta _1\theta _2),\,\, R=2E[{\tilde{\beta }}](3\theta _1-\tau \theta _2),\,\,\\ S& {}=\, 2E[{\tilde{\beta }}](2+\tau ^2+\theta _2^2),\,\,T=2E[{\tilde{\beta }}](3\theta _2-\tau \theta _1),\,\,U=2E[{\tilde{\beta }}](2+\theta _1^2+\theta _2^2),\\ B_1& {}= 3E[{\tilde{a}}_1]-2\tau E[{\tilde{a}}_2]+2\theta _1E[{\tilde{a}}_3]+E[{\tilde{c}}_1{\tilde{\beta }}]-\tau E[{\tilde{c}}_2{\tilde{\beta }}]+\theta _1E[{\tilde{c}}_3{\tilde{\beta }}],\,\,\\ B_2& {}=\, 3E[{\tilde{a}}_2]-2\tau E[{\tilde{a}}_1]+2\theta _2E[{\tilde{a}}_3]-\tau E[{\tilde{c}}_1{\tilde{\beta }}]+E[{\tilde{c}}_2{\tilde{\beta }}]+\theta _2E[{\tilde{c}}_3{\tilde{\beta }}],\,\,\\ B_3& {}=\, 3E[{\tilde{a}}_3]+2\theta _1E[{\tilde{a}}_1]+2\theta _2E[{\tilde{a}}_2]+\theta _1E[{\tilde{c}}_1{\tilde{\beta }}]+\theta _2E[{\tilde{c}}_2{\tilde{\beta }}]+E[{\tilde{c}}_3{\tilde{\beta }}]. \end{aligned}$$
The above retailer’s expected profit expression in RS approach is concave if
$$\begin{aligned} \tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2<1\,and\,\tau ^2+\tau \theta _1\theta _2+\theta _1^2+\theta _2^2<4. \end{aligned}$$
At the extreme point, we have
$$\begin{aligned} P s^*_1-Q s^*_2+R s^*_3=B_1,\,\,\,-Q s^*_1+S s^*_2+T s^*_3=B_2\,\,\,{\mathrm{and}}\,\,\,R s^*_1+T s^*_2+U s^*_3=B_3 \end{aligned}$$
(43)
Solving the system of linear Eqs. given in (43), we have
$$\begin{aligned} s^*_1= \,& {} \frac{B_1(T^2-SU)-B_2(RT+QU)+B_3(QT+RS)}{(UQ^2+2QRT+SR^2+PT^2-PSU)}\nonumber \\ s^*_2=\, & {} \frac{-B_1(RT+QU)+B_2(R^2-PU)+B_3(QR+PT)}{(UQ^2+2QRT+SR^2+PT^2-PSU)}\nonumber \\ s^*_3= \,& {} \frac{B_1(QT+RS)+B_2(QR+PT)+B_3(Q^2-PS)}{(UQ^2+2QRT+SR^2+PT^2-PSU)} \end{aligned}$$
(44)
From the constraints \({{\textit{Pos}}}(\{\tilde{a_1}\ge {\tilde{\beta }} s^*_1-\tau {\tilde{\beta }} s^*_2+\theta _1{\tilde{\beta }} s^*_3\})>\eta _1\), \({{\textit{Pos}}}(\{\tilde{a_2}\ge -\tau {\tilde{\beta }} s^*_1 +{\tilde{\beta }} s^*_2+\theta _2{\tilde{\beta }} s^*_3\})>\eta _1\) and \({{\textit{Pos}}}(\{\tilde{a_3}\ge \theta _1{\tilde{\beta }} s^*_1+\theta _2{\tilde{\beta }} s^*_2+{\tilde{\beta }} s^*_3\})>\eta _1\) as similar as “Appendix 1”, we have
$$\begin{aligned} {\tilde{a}}^{R}_{1\eta _1}>{\tilde{\beta }}^{L}_{\eta _1} s^*_1-\tau {\tilde{\beta }}^{R}_{\eta _1} s^*_2+\theta _1{\tilde{\beta }}^{L}_{\eta _1} s^*_3,\,\, {\tilde{a}}^{R}_{2\eta _1}>-\tau {\tilde{\beta }}^{R}_{\eta _1} s^*_1+{\tilde{\beta }}^{L}_{\eta _1} s^*_2 +\theta _2{\tilde{\beta }}^{L}_{\eta _1} s^*_3,\,\,{\tilde{a}}^{R}_{3\eta _1}>\theta _1{\tilde{\beta }}^{L}_{\eta _1} s^*_1+\theta _2{\tilde{\beta }}^{L}_{\eta _1} s^*_2+{\tilde{\beta }}^{L}_{\eta _1} s^*_3. \end{aligned}$$
Appendix 4
The optimum decisions of three manufacturers and one retailer are derived in RS and MS approaches and given in Eqs. (39), (40), (41), (26), (27) and (28). For Nash-equilibrium strategies, solving these six equations, we have
$$\begin{aligned} p^*_1=\, & {} -\frac{\left( 9-\theta _2^2\right) \left( E\left[ {\tilde{a}}_1\right] +E\left[ {\tilde{c}}_1{\tilde{\beta }}\right] \right) +\left( 3\tau +\theta _1\theta _2\right) \left( E\left[ {\tilde{a}}_2\right] +E\left[ {\tilde{c}}_2{\tilde{\beta }}\right] \right) -\left( 3\theta _1+\tau \theta _2\right) \left( E\left[ {\tilde{a}}_3\right] +E\left[ {\tilde{c}}_3{\tilde{\beta }}\right] \right) }{E\left[ {\tilde{\beta }}\right] \left( 3\tau ^2+2\tau \theta _1\theta _2+3\theta _1^2+3\theta _2^2-27\right) } \end{aligned}$$
(45)
$$\begin{aligned} p^*_2=\, & {} -\frac{\left( 3\tau +\theta _1\theta _2\right) \left( E\left[ {\tilde{a}}_1\right] +E\left[ {\tilde{c}}_1{\tilde{\beta }}\right] \right) +\left( 9-\theta _1^2\right) \left( E\left[ {\tilde{a}}_2\right] +E\left[ {\tilde{c}}_2{\tilde{\beta }}\right] \right) -\left( 3\theta _2+\tau \theta _1\right) \left( E[{\tilde{a}}_3]+E[{\tilde{c}}_3{\tilde{\beta }}]\right) }{E[{\tilde{\beta }}]\left( 3\tau ^2+2\tau \theta _1\theta _2+3\theta _1^2+3\theta _2^2-27\right) }\end{aligned}$$
(46)
$$\begin{aligned} p^*_3=\, & {} \frac{\left( 3\theta _1+\tau \theta _2\right) \left( E[{\tilde{a}}_1]+E[{\tilde{c}}_1{\tilde{\beta }}]\right) +\left( 3\theta _2+\tau \theta _1\right) \left( E[{\tilde{a}}_2]+E[{\tilde{c}}_2{\tilde{\beta }}]\right) -\left( 9-\tau ^2\right) \left( E[{\tilde{a}}_3]+E[{\tilde{c}}_3{\tilde{\beta }}]\right) }{E[{\tilde{\beta }}]\left( 3\tau ^2+2\tau \theta _1\theta _2+3\theta _1^2+3\theta _2^2-27\right) }\end{aligned}$$
(47)
$$\begin{aligned} s^*_1= \,& {} \frac{p^*_1}{2}-\frac{\left( 1-\theta _2^2\right) E[{\tilde{a}}_1]+\left( \tau +\theta _1\theta _2\right) E[{\tilde{a}}_2]-\left( \theta _1+\tau \theta _2\right) E[{\tilde{a}}_3]}{2E[{\tilde{\beta }}]\left( \tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1\right) }\end{aligned}$$
(48)
$$\begin{aligned} s^*_2= \,& {} \frac{p^*_2}{2}-\frac{\left( \tau +\theta _1\theta _2\right) E[{\tilde{a}}_1]+\left( 1-\theta _1^2\right) E[{\tilde{a}}_2]-\left( \theta _2+\tau \theta _1\right) E[{\tilde{a}}_3]}{2E[{\tilde{\beta }}]\left( \tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1\right) }\end{aligned}$$
(49)
$$\begin{aligned} s^*_3=\, & {} \frac{p^*_3}{2}+\frac{\left( \theta _1+\tau \theta _2\right) E[{\tilde{a}}_1]+\left( \theta _2+\tau \theta _1\right) E[{\tilde{a}}_2]-\left( 1-\tau ^2\right) E[{\tilde{a}}_3]}{2E[{\tilde{\beta }}]\left( \tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1\right) } \end{aligned}$$
(50)
