## 4 Computational Study and Analysis
### Evolution Results and Analysis  
Suppose a number of retail supermarkets sell eight kinds of products, each of which is supplied by a single manufacturer. Eight different simulation experiments will be carried out. The evolution process and results are shown from Figures 2-7, simulating six types formations of evolution results. Among the six pictures, the real-line curve represents the evolution trend of the manufacturer's emission reduction strategy, and the virtual-line indicates the evolution trend of the retailer's low-carbon \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Production** & \(\mathbf{u_{M}}\) & \(\mathbf{u_{R}}\) & \(\mathbf{c_{M}}\) & \(\mathbf{c_{R}}\) & \(\mathbf{q_{1}}\) & \(\mathbf{q_{2}}\) & \(\mathbf{q_{3}}\) \\ \hline A\({}_{1}\) & 0.1 & 0.08 & 0.5 & 0.45 & 8 & 1 & 4 \\ A\({}_{2}\) & 0.2 & 0.16 & 0.2 & 0.8 & 7 & 5 & 3 \\ A\({}_{3}\) & 0.2 & 0.16 & 0.2 & 0.35 & 8 & 6 & 4 \\ A\({}_{4}\) & 0.11 & 0.21 & 0.9 & 0.45 & 8 & 1 & 4 \\ A\({}_{5}\) & 0.4 & 0.21 & 0.5 & 0.15 & 8 & 2 & 7 \\ A\({}_{6}\) & 0.2 & 0.21 & 0.5 & 0.65 & 8 & 4 & 2 \\ A\({}_{7}\) & 0.2 & 0.21 & 0.5 & 0.65 & 8 & 2 & 2 \\ A\({}_{8}\) & 0.2 & 0.21 & 0.5 & 0.65 & 8 & 5 & 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of Simulation Experiment. promotion strategy. The results of the simulation experiment verify those managerial insights that are mentioned in Section 3. **(1) The system evolutionary stability strategy \(E_{1}(0,0)\)**  
In the process of evolution concerning the first category (as shown in Figure 2), we conduct 100 times simulation (procurement procedures) for manufacturers and retailers, which involved A\({}_{1}\), and dynamic replicator system converges to point \(E_{1}(0,0)\). However, we set different initial conditions (\(x=0.13\) and \(y=0.15\) in Figure 2a and \(x=0.83\) and \(y=0.85\) in Figure 2b). The system varies in the convergent process. For example, most of manufacturers tend to choose emission reduction in Figure 2b in the first 10 times simulation (\(x\) tended to 1), and then fell rapidly, then converged to \(E_{1}(0,0)\). Even if the system set high initial value (\(x=0.83\) and \(y=0.85\)) in the beginning, two sides still fail to cooperate eventually. The main reason is that, when two parties reduced emission individually, the profit of them reducing emission is less than that they quit reducing emission; in addition, when two sides cooperate on emission reduction, the retailers' profit emission is more than that they choose free riding. However, the manufacturers' profit is less than that under free riding when they cooperate to reduce emission. Therefore, among several games, retailers choose to cooperate but manufacturers adopt free riding, which induces retailers change their strategies to quit cooperating. Finally, ESS is adopted by two parties is non-cooperative emission. **(2) The system evolutionary stable strategy \(E_{2}(1,0)\)**  
In the process of evolution concerning the second category (as shown in Figure 3), we conduct 100 times simulation (procurement procedures) for manufacturers and retailers, which involved A\({}_{2}\) and A\({}_{3}\), respectively, and the dynamic replicator system converges to \(E_{2}(1,0)\). We set the same initial conditions (\(x=0.15\) and \(y=0.63\)) in Figure 3a,b, which means the proportion of manufacturers choosing high-level carbon emission reduction strategy is lower, but the proportion of retailers choosing high-level low carbon promotion strategy is higher at the beginning. Finally, they all converge to \(E_{2}(1,0)\). Furthermore, the system varies in the convergence process. In Figure 3b, retailers tended to choose cooperative emission reduction in the first 5 times simulation, and then gradually shifted till converged to \(E_{2}(1,0)\).