# Unfair and Risky? Profit Allocation in Closed-Loop Supply Chains by Cooperative Game Approaches
## Appendix C Proof of Proposition 3
Plugging demand functions \(q_{n}^{ML}\big{(}p_{n}^{ML},p_{r}^{ML}\big{)}\) and \(q_{r}^{ML}\big{(}p_{n}^{ML},p_{r}^{ML}\big{)}\) into ML's utility function and taking partial derivatives, we have \(\frac{\partial^{2}u_{n}^{ML}}{\partial p_{n}^{ML}}=\frac{2}{\delta-1}<0\), \(\frac{\partial^{2}u_{n}^{ML}}{\partial p_{n}^{ML}}=\frac{2}{\delta-1}-\frac{2 }{\delta}<0\), and \(\frac{\partial^{2}u_{n}^{ML}}{\partial p_{n}^{ML}\partial p_{n}^{ML}}=\frac{2 }{1-\delta}>0\). So \(\frac{\partial^{2}u_{n}^{ML}}{\partial p_{n}^{ML}}\frac{\partial^{2}u_{n}^{ML }}{\partial p_{n}^{ML}}-\left(\frac{\partial^{2}u_{n}^{ML}}{\partial p_{n}^{ ML}\partial p_{n}^{ML}}\right)^{2}=\frac{4(1-\delta)}{(1-\delta)^{2}\delta}>0\) implies that ML's utility function is joint concave in \(p_{n}^{ML}\) and \(p_{r}^{ML}\), and \(u_{ML}^{ML}\) has a unique optimal solution. With first-order conditions \(\frac{\partial u_{n}^{ML}}{\partial p_{n}^{ML}}=0\) and \(\frac{\partial u_{n}^{ML}}{\partial p_{r}^{ML}}=0\), ML's optimal responses are obtained as \(p_{n}^{*ML}=\frac{c_{n}+1}{2}\), \(p_{r}^{*ML}=\frac{c_{r}+\lambda+\delta}{\delta+2}\). By substituting \(p_{n}^{F}\big{(}w_{n}^{F},\ w_{r}^{F}\big{)}\), \(p_{r}^{F}\big{(}w_{n}^{F},\ w_{r}^{F}\big{)}\), \(p_{n}^{ML}\) and \(p_{r}^{ML}\) into demand functions \(q_{n}^{F}\big{(}p_{n}^{F},p_{r}^{F}\big{)}\), \(q_{r}^{F}\big{(}p_{n}^{F},p_{r}^{F}\big{)}\), \(q_{n}^{ML}\big{(}p_{n}^{ML},p_{r}^{ML}\big{)}\) and \(q_{r}^{ML}\big{(}p_{n}^{ML},p_{r}^{ML}\big{)}\), the optimal quantities are obtained as M's wholesale prices to F. By plugging the optimal quantities into ML's utility function with the first-order conditions \(\frac{\partial u_{n}^{ML}}{\partial w_{n}^{ML}}=0\) and \(\frac{\partial u_{n}^{ML}}{\partial w_{r}^{F}}=0\), ML's optimal decisions \(w_{n}^{*F}=\frac{\lambda+c_{n}+3\lambda c_{n}+1}{4\lambda+2}\) and \(w_{r}^{*F}=\frac{c_{r}+\lambda+\delta+3\lambda c_{n}+3\lambda A+\lambda\delta}{ 4\lambda+2}\) are obtained. \(\square\)  
Appendix E Proof of Proposition 5. In Model MF, alliance MF and L compose a two-echelon model with MF being the leader and L the follower in a Stackelberg game. By plugging demand functions \(q_{n}^{L}\big{(}p_{n}^{L},p_{r}^{L}\big{)}\) and \(q_{r}^{L}\big{(}p_{n}^{L},p_{r}^{L}\big{)}\) into L's utility function and calculating partial derivatives, we get \(\frac{\partial^{2}u_{n}^{MF}}{\partial p_{n}^{L^{2}}}=-\eta\sigma^{2}-\frac{2}{1- \delta}<0\), \(\frac{\partial^{2}u_{n}^{MF}}{\partial p_{n}^{L^{2}}}=-\eta\sigma^{2}-\frac{2}{1- \delta}-\frac{2}{\delta}<0\), and \(\frac{\partial^{2}u_{n}^{MF}}{\partial p_{n}^{L}\partial p_{r}^{L}}=\frac{2}{1- \delta}-\eta\sigma^{2}\rho>0\). Then \(\frac{\partial^{2}u_{n}^{MF}}{\partial p_{n}^{L^{2}}}\frac{\partial^{2}u_{n}^{MF} }{\partial p_{n}^{L^{2}}}-\left(\frac{\partial^{2}u_{n}^{MF}}{\partial p_{n}^{L} \partial p_{r}^{L}}\right)^{2}=\left(\frac{2}{1-\delta}+\eta\sigma^{2}\right)* \left(\frac{2}{1-\delta}+\eta\sigma^{2}+\frac{2}{\delta}\right)-\left(\frac{2}{1- \delta}-\eta\sigma^{2}\rho\right)^{2}>0\) implies that L's utility function \(u_{L}^{MF}\) is joint concave in \(p_{n}^{L}\) and \(p_{r}^{L}\) with a unique optimal solution.