# Unfair and Risky? Profit Allocation in Closed-Loop Supply Chains by Cooperative Game Approaches
## Appendix C Proof of Proposition 3  
In the LF model, the alliance LF decides the sales prices to maximize the coalition utility. By substituting demand functions \(q_{n}^{LF}\left(p_{n}^{LF},p_{r}^{LF}\right)\) and \(q_{r}^{LF}\left(p_{n}^{LF},p_{r}^{LF}\right)\) into LF's utility function \(u_{LF}^{LF}\) and taking partial derivatives, we have \(\frac{\partial^{2}u_{LF}^{LF}}{\partial p_{n}^{LF}}=-\eta\sigma^{2}-\frac{2+2 \lambda}{1-\delta}<0\), \(\frac{\partial^{2}u_{LF}^{LF}}{\partial p_{r}^{LF}}=-\eta\sigma^{2}-\frac{2}{1- \delta}-\frac{2}{\delta}-\lambda\left(\frac{2}{1-\delta}+\frac{2}{\delta} \right)<0\), and \(\frac{\partial^{2}u_{LF}^{LF}}{\partial p_{n}^{LF}}=\frac{2+2\lambda}{1-\delta}- \eta\rho\sigma^{2}>0\). Then \(\frac{\partial^{2}u_{LF}^{LF}}{\partial p_{n}^{LF}}\frac{\partial^{2}u_{LF}^{ LF}}{\partial p_{r}^{LF}}-\left(\frac{\partial^{2}u_{LF}^{LF}}{\partial p_{n}^{LF}} \frac{\partial^{2}u_{LF}^{LF}}{\partial p_{r}^{LF}}-\left(\frac{\partial^{2}u_{LF}^{ LF}}{\partial p_{n}^{LF}}\frac{\partial^{2}u_{LF}^{LF}}{\partial p_{r}^{LF}} \right)^{2}=\frac{\left(-\eta\sigma^{2}\left(1-\delta\right)-2-2\lambda\right) \left(-\eta\sigma^{2}\delta\left(1-\delta\right)-2\delta-2\left(1-\delta \right)-2\lambda\right)-\delta\left(-2\lambda-\eta\sigma\sigma^{2}\left(1-\delta \right)\right)^{2}}{\delta\left(1-\delta\right)^{2}}\)\(>0\) implies that LF's utility function \(u_{LF}^{LF}\) is joint concave in \(p_{n}^{LF}\) and \(p_{LF}^{LF}\), and \(u_{LF}^{LF}\) has a unique optimal solution. With first-order conditions \(\frac{\partial u_{LF}^{LF}}{\partial p_{n}^{LF}}=0\) and \(\frac{\partial u_{LF}^{LF}}{\partial p_{r}^{LF}}=0\), LF's optimal response functions are obtained as  
\[\begin{array}{l}p_{n}^{LF}\left(u_{n}^{LF},u_{r}^{LF}\right)=\frac{Ku_{n}^{ LF}+\left(\eta\sigma^{2}\rho+\eta\sigma^{2}\delta\right)w_{n}^{LF}-\eta\sigma^{2} \lambda\delta^{2}+\eta\sigma^{2}\rho\lambda A-\eta\sigma^{2}\lambda c_{n} \delta+\eta\sigma^{2}\lambda A\delta-\eta\sigma^{2}\rho\lambda c_{n}\delta+4 \lambda-2\lambda c_{n}-2\lambda c_{n}-2\lambda^{2}c_{n}+2\eta\sigma^{2}\delta +\eta\sigma^{2}\lambda\delta+2\eta\sigma^{2}\lambda\delta+\eta\sigma^{2}  
Appendix D Proof of Proposition 4.1. In Model ML, coalition ML and F constitute a two-echelon Stackelberg game model with ML being the leader and F the follower. Plugging demand functions \(q_{n}^{F}\big{(}p_{n}^{F},p_{r}^{F}\big{)}\) and \(q_{r}^{F}\big{(}p_{n}^{F},p_{r}^{F}\big{)}\) into F's utility function and taking partial derivatives, we have \(\frac{\partial^{2}u_{n}^{ML}}{\partial p_{n}^{2}}=\frac{2\lambda+2}{\delta-1}< 0\), \(\frac{\partial^{2}u_{n}^{ML}}{\partial p_{r}^{2}}=\left(\frac{2}{\delta-1}- \frac{2}{\delta}\right)(\lambda+1)<0\), and \(\frac{\partial^{2}u_{n}^{ML}}{\partial p_{n}^{2}\partial p_{r}^{F}}=\frac{2 \lambda+2}{1-\delta}>0\). This implies that \(\frac{\partial^{2}u_{n}^{ML}}{\partial p_{n}^{2}}\frac{\partial^{2}u_{n}^{ML} }{\partial p_{r}^{2}}-\left(\frac{\partial^{2}u_{n}^{ML}}{\partial p_{n}^{2} \partial p_{r}^{F}}\right)^{2}=\frac{4(\lambda+1)^{2}(1-\delta)}{\delta(1- \delta)^{2}}>0\). Thus, F's utility function \(u_{F}^{ML}\) is strictly joint concave in \(p_{n}^{F}\) and \(p_{r}^{F}\), and it has a unique optimal solution. With first-order conditions \(\frac{\partial u_{n}^{ML}}{\partial p_{n}^{2}}=0\) and \(\frac{\partial u_{n}^{ML}}{\partial p_{r}^{F}}=0\), F's optimal response functions are obtained \(p_{n}^{F}\big{(}w_{n}^{F},\ w_{r}^{F}\big{)}=\frac{\lambda+w_{r}^{F}-\lambda c _{n}+2\lambda w_{r}^{F}+2}{2\lambda+2}\), \(p_{r}^{F}\big{(}w_{r}^{F},\ w_{r}^{F}\big{)}=\frac{\delta+w_{r}^{F}-\lambda c _{r}-\lambda\lambda+\lambda+\delta+2\lambda w_{r}^{F}}{2\lambda+2}\).