## 2 Prerequisites and Assumptions  
We considered eight different scenarios, namely Scenarios \(ij\), \(i\in\{m,r\}\), which signifies MS and RS games; \(j\in\{C,RE,T,N\}\), which refers to the models where consumers receive a subsidy (C), the manufacturer receives a subsidy on the investment effort on recycling (RE), and the manufacturer receives a subsidy on the total R&D investment (T); and the benchmark decision model where the government organizations do not provide a subsidy (N), respectively. Therefore, the first index represents the game structure, and the second one represents the subsidy policy. The following notations presented in Table 2 are used to differentiate the decision and auxiliary variables in different scenarios:  
The following assumptions are made to establish proposed models:  
1. Similar to [PERSON] et al. [60]; [PERSON] et al. [14]; [PERSON] and [PERSON] [61], the market demand is linearly dependent on the retail price and GL, and its functional form is \(D_{i}^{j}=a-p_{i}^{j}+\beta\theta_{i}^{j}\), where \(a\) represents potential intrinsic demand, and \(\beta(>0)\) represents GL sensitivity. Therefore, higher value for \(a\) means the consumer has the better perception about the product. For analytical simplicity, the coefficient of price sensitivity is normalized with the unit [15]. 2. It is assumed that re-manufacturing cost is less compared to manufacturing cost, i.e., \(c_{r}<c_{m}\)[6; 57]. The manufacturer invests \(CL(\tau_{i}^{j},\alpha)=\alpha\tau_{i}^{j}D_{i}^{j}+\kappa\tau_{i}^{j}\) to collect used products, where \(\kappa>0\) represents the scaling parameter, and \(\delta\alpha>0\) represents the monetary benefit's for the consumers for returning used products. If \(\alpha=0\), this assumption is similar to [PERSON] et al. [28], [PERSON] et al. [53], and [PERSON] and [PERSON] [57]. \(\tau_{i}^{j}\) represents the collection rate \((0\leq\tau_{i}^{j}\leq 1)\). While optimizing objective functions, all members have access to the same information [51; 59]. A portion \(\delta\) (\(0\leq\delta\leq 1\)) of the collected used products converts into new one [57]. 3. The manufacturer bears extra cost for green technology innovation. In a recent study by [PERSON] et al. [62], it was found that technological innovation have significant effects on regional industrial eco-efficiency. In this study, we assume that the manufacturer produces MDIGPs, and corresponding per unit and total R&D investment costs are considered as \(\lambda_{1}\theta\) and \(\lambda_{2}\theta^{2}\), respectively. Therefore, \(\lambda_{1}\) and \(\lambda_{2}\) represents the efficiency of the manufacturer on per unit investment and investment in R&D, respectively. If \(\lambda_{1}=0\), then the manufacturer invests in producing DIGPs [18; 19]. If \(\lambda_{2}=0\), then the product converts to MIGPs [63]. The fixed cost for the \begin{table}
\begin{tabular}{l l} \hline \(w_{i}^{j}\) & wholesale price of the new/re-manufacturing product \\ \(p_{i}^{j}\) & market price \\ \(\tau_{i}^{j}\) & collection rate of used products \\ \(\theta_{i}^{j}\) & greening level \\ \(\rho_{i}^{j}\) & per unit subsidy received by the consumer from the government, \((\rho_{i}^{j}<p_{i}^{j})\) \\ \(\eta_{i}^{j}\) & subsidy received by the manufacturer on investment to improve recycling, \((0\leq\eta_{i}^{j}\leq 1)\) \\ \(\mu_{i}^{j}\) & fraction of subsidy received by the manufacturer on investment to improve GL, \((0\leq\eta_{i}^{j}\leq 1)\) \\ \(\pi_{ki}^{j}\) & member k’s profit, \(k\in\{m,r\}\) \\ \(\pi_{gi}^{j}\) & SW of government \\ \(Q_{i}^{j}\) & sales volume \\ \hline \end{tabular}
\end{table} 
Table 2: Decision and auxiliary variables. retailer and manufacturer are normalized to zero [14; 15]. The government organizations provide a R&D subsidy on the total investment. As noted by [PERSON] et al. [14], for MIGPs, the variable cost is directly proportional with the product quality, and it might not possible to recover the cost for the manufacturers.