# Unfair and Risky? Profit Allocation in Closed-Loop Supply Chains by Cooperative Game Approaches
## 4 The Equilibrium Analysis
### Analytical Comparison of Resulting Equilibriums  
The resulting equilibriums in Propositions 1-5 are assessed comparatively and the conclusions are drawn as the following. **Proposition 6**. : _The wholesale pricing of the five models satisfy_  
\[\left( \begin{matrix}{{w_{n}^{*L}}}\\ \end{matrix} \right)^{DC}=\left({{w_{n}^{*L}}}\right)^{MF},\left({{w_{r}^{*L}}} \right)^{DC}=\left({{w_{r}^{*L}}}\right)^{MF},\left({{w_{n}^{*F}}}\right)^{DC} =\left({{w_{n}^{*F}}}\right)^{ML},\] \[\left({{w_{r}^{*F}}}\right)^{DC}=\left({{w_{r}^{*F}}}\right)^{ML}, \text{ and }\frac{{\partial\left({{w_{n}^{*L}}}\right)}^{i}}{{\partial t}}=\frac{{ \partial\left({{w_{r}^{*L}}}\right)}^{i}}{{\partial t}}=0\text{ where }i\in\{DC, MF\}.\] \[\left( \begin{matrix}{{2}}&\text{If }\ A+\delta<1,\left({{w_{n}^{*L}}} \right)^{i}>\ \left({{w_{r}^{*L}}}\right)^{i}\text{ where }i\in\{DC, MF\}.\\ \end{matrix} \right.\] \[\left( \begin{matrix}{{3}}&\text{If }\ \left({{3}}&\text{{{{1}}}} \right)({{c_{r}}}+A-{{c_{n}}})+\lambda(\delta-1)+\delta<1,\left({{w_{n}^{*F}}} \right)^{i}>\ \left({{w_{r}^{*F}}}\right)^{i}\text{ where }i\in\{DC, ML\}.\\ \end{matrix} \right.\]  
Proposition 6(1) demonstrates that the wholesale price from M to L in Model DC equals that in Model MF. It also presents that the wholesale price from M to F in Model DC is equivalent to that in Model ML. This observation is explainable that M makes the same decision of wholesale price to the downstream single party (i.e., L or F). Thus we have \(\left({{w_{n}^{*L}}}\right)^{DC}=\left({{w_{n}^{*L}}}\right)^{MF},\quad \left({{w_{r}^{*L}}}\right)^{DC}=\left({{w_{r}^{*L}}}\right)^{MF},\quad\left( {{w_{n}^{*F}}}\right)^{DC}=\left({{w_{n}^{*F}}}\right)^{ML},\) and \(\left({{w_{r}^{*F}}}\right)^{DC}=\left({{w_{r}^{*F}}}\right)^{ML}\). In addition, M considers L's risk aversion only when [PERSON] and [PERSON] form an alliance. Therefore, M makes decision of wholesale prices independently of [PERSON]'s risk aversion parameter \(\eta\) in the models of DC and MF. Therefore, we have \(\frac{{\partial\left({{w_{n}^{*L}}}\right)}^{i}}{{\partial\eta}}=\frac{{ \partial\left({{w_{r}^{*L}}}\right)}^{i}}{{\partial\eta}}=0,\) where \(i\in\{DC, MF\}.\)  
Proposition 6(2) is a comparison of wholesale prices for new and remanufactured products in the DC and MF models. When \(A+\delta<1\), L has higher willingness to pay for new products than remanufactured products. Proposition 6(3) compares the wholesale prices of new and remanufactured products of the DC and ML models. When \((3\lambda+1)({{c_{r}}}+A-{{c_{n}}})+\lambda(\delta-1)+\delta<1\), F has lower willingness to pay for remanufactured products. **Proposition 7**. : _The retail pricing of new and remanufactured products in the five models satisfy_  
\[ \begin{matrix}1&p_{n}^{*ML}=p_{n}^{*MF}=p_{n}^{*LC}=p_{n}^{*FC},\ p_{r}^{*ML}=p_{r}^{*MF}=p_{r}^{L*CC}=p_{r}^{F*CC}, \text{ and }\frac{{\partial p_{r}^{*i}}}{{\partial t}}=\frac{{\partial p_{r}^{*i}}}{{ \partial\delta}}=\frac{{\partial p_{r}^{*i}}}{{\partial t}}=\frac{{\partial p _{r}^{*i}}}{{\partial t}}=0\text{ where }i\in\{ML,MF,CC\}.\\ 2&p_{n}^{*CC}<p_{n}^{*LC},\ p_{r}^{*CC}<p_{r}^{*DC}\\ 3&\text{ If }A+\delta<1,\ p_{n}^{*i}>p_{r}^{*i}\text{ where }i\in\{ML,MF,CC\}.\\ \end{matrix} \]  
Proposition 7(1) shows that the retail prices of the ML coalition, MF coalition, and centralized model are the same. F's fairness concerns and L's risk aversion parameter do not affect the retail prices of the ML, MF, and MLF alliances. This is clear given that L or F partners with upstream agent M in the three models. Proposition 7(2) concludes that the retail price of L in the CC model is always lower than that in the DC model. This is due to customers having higher willingness to purchase products from L than MLF. Proposition 7(3) is a comparison of retail prices for new and remanufactured products in the three models. When \(A+\delta<1\), consumers have higher willingness to buy new products than remanufactured products. **Proposition 8**.