# Unfair and Risky? Profit Allocation in Closed-Loop Supply Chains by Cooperative Game Approaches
## 3 Problem Description and Assumptions  
This section describes a two-echelon supply chain model consisting of a risk-neutral manufacturer (M) and two competing retailers (L and F) who play a Stackelberg game. It is assumed that Retailer L has a higher market share than Retailer F, so that customers have higher willingness to purchase products from L than F. Given Retailer F's relatively weaker position and the wide recognition that disadvantaged agents often care about fairness issues [31,32], this research assumes that Retailer F has a fairness concern with the upper manufacturer. We assume Retailer L is risk-averse and attempts to avoid risks. The sales channels provided by the two retailers are available to all customers. Customers are risk neutral and make decisions to maximize their utility, and we consider one-period interactions among CLSC members. To derive characteristic functions of the cooperative game, all potential alliances and the corresponding equilibriums are investigated. Five models are presented in Figure 1: the centralized model, CC, with a central decision maker, the decentralized model, DC, where agents make individual decisions independently, and three partial-cooperation models, LF, ML, and MF, where three partial alliances form and make centralized decisions within the respective coalition. On the basis of the problem description, this paper employs the following notation throughout the research as shown in Table 2. \begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & **Definition** \\ \hline \(c_{n}\), \(c_{r}\) & Unit production cost for a new or remanufactured product \\ \(w_{r}^{L}\), \(w_{r}^{L}\) & Unit wholesale price for a new or remanufactured product offered by M to L \\ \(w_{r}^{L}\), \(w_{r}^{L}\) & Unit wholesale price for a new or remanufactured product offered by M to F \\ \(p_{p}^{L}\), \(p_{r}^{L}\) & Unit retail price for a new or remanufactured product offered by L \\ \(p_{r}\), \(p_{r}^{L}\) & Unit retail price for a new or remanufactured product offered by F \\ \(q_{m}\), \(q_{r}\) & Production quantity for new or remanufactured products \\ \(q_{m}^{L}\), \(q_{r}^{L}\) & Quantity of new or remanufactured products transferred from M to L \\ \(q_{r}^{L}\), \(q_{r}^{L}\) & Quantity of new or remanufactured products transferred from M to F \\ \(t_{m}^{L}\), \(t_{r}^{L}\) & Utility of a consumer receiving from L for a new or remanufactured product \\ \(t_{n}^{L}\), \(t_{r}^{F}\) & Utility of a consumer receiving from F for a new or remanufactured product \\ \(A\) & Exogenous unit cost for recycling a preowned product \\ \(\lambda\) & F’s fairness concern parameter \\ \(\eta\) & L’s risk aversion parameter, where \(\eta\in(0,1)\) \\ \(\theta\) & Consumers’ willingness-to-pay for a new product \\ \(\delta\) & Consumers’ value deduction for a remanufactured product, \(\delta\in[0,1)\) \\  & The correlation coefficient between new products and \\ \(\rho\) & remanufactured products \\ \(\sigma^{2}\) & The variance of demand uncertainty \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Variables and parameters. Figure 1: The cooperative, non-cooperative, and partial-cooperative models of the two-echelon CLSC consisting of one manufacturer and two retailers are: **(a)** Model CC; **(b)** Model DC; **(c)** Model LF, **(d)** Model ML; and **(e)** Model MF. In this CLSC, consumers' willingness to pay for a new product is assumed to be \(\theta\), which is a uniform distribution from 0 to 1. For a remanufactured product, consumers' willingness to pay is a portion \(\delta\) of \(\theta\) with \(\delta\in[0,1)\). The utility of a consumer receiving from Retailer L for a new product is \(t_{n}^{L}(\theta)=\theta-p_{n}^{L}\), and the utility of a consumer receiving from Retailer L for a remanufactured product is \(t_{r}^{L}(\theta)=\delta\theta-p_{r}^{L}\).