## 3 Evolutionary Game Model for Carbon Emission Reduction
### Mathematical Models  
**Step 1**: Establishing a pay-off matrix  
Denote \(x\) (\(0\leq x\leq 1\)) as the proportion of manufacturers choosing high-level emission reduction strategy and otherwise \(1-x\) as those choosing low-level strategy. Similarly, denote \(y\) (\(0\leq y\leq 1\)) the proportion of retailers choosing high-level strategy on promoting low-carbon products, and \(1-y\) the proportion of retailers choosing low-level strategy. Under above assumptions, the payment matrix of both parties is as shown in Table 1. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Manufacturers**} & \multicolumn{2}{c}{**Retailers**} \\ \cline{2-3}  & **High-Level Promotion** & **Low-Level Promotion** \\  & **on Low Carbon (H)** & **on Low Carbon (L)** \\ \hline High-level emission & \(r_{M}+u_{M}q_{1}-c_{M},r_{R}+u_{R}q_{1}-c_{R}\) & \(r_{M}+u_{M}q_{2}-c_{M},r_{R}+u_{R}q_{2}\) \\ reduction investment (\(H\)) & & \(r_{M}+u_{M}q_{3},r_{R}+u_{R}q_{3}-c_{R}\) & \(r_{M},r_{R}\) \\ \hline Low-level emission & \(r_{M}+u_{M}q_{3},r_{R}+u_{R}q_{3}-c_{R}\) & \(r_{M},r_{R}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: A pay-off matrix of two players. According to Table 1, we express the expected profits for manufacturers' strategies of high-level emission reduction and low-level emission reduction and their average profits, and \(U_{MH}\), \(U_{ML}\) and \(\overline{U_{M}}\) as follows, respectively. \[U_{MH}=(r_{M}+u_{M}q_{1}-c_{M})y+(r_{M}+u_{M}q_{2}-c_{M})(1-y) \tag{1}\]  
\[U_{ML}=(r_{M}+u_{M}q_{3})y+r_{M}(1-y) \tag{2}\]  
\[\overline{U}_{M}=xU_{MH}+(1-x)U_{ML} \tag{3}\]  
For manufacturers, when they choose high-level emission reduction strategy, the dynamic change rate for the proportion of manufacturers choosing high-level emission reduction strategy can be expressed as the following dynamic differential equation:  
\[\frac{dx}{dt}=x(U_{MH}-\overline{U_{M}}) \tag{4}\]  
Substituting Equations (1)-(3) into (4), we have:  
\[\frac{dx}{dt}=x(1-x)[u_{M}(q_{1}-q_{2}-q_{3})y+u_{M}q_{2}-c_{M}] \tag{5}\]  
The differential Equation (5) is called the replicator dynamic equation. The speed of rate change for the proportion of manufacturers choosing high level emissions reduction strategy is positively related to both the proportion and the difference between the expected profit choosing this strategy and the average profit. Similarly, retailers' expected profits incurred respectively by high-level and low-level promotion on low carbon, the average profit are expressed as \(U_{RH}\), \(U_{RL}\), and \(\overline{U_{R}}\), respectively. \[U_{RH}=(r_{R}+u_{R}q_{1}-c_{R})x+(r_{R}+u_{R}q_{3}-c_{R})(1-x) \tag{6}\]  
\[U_{RL}=(r_{R}+u_{R}q_{2})x+r_{R}(1-x) \tag{7}\]  
\[\overline{U_{R}}=yU_{RH}+(1-y)U_{RL} \tag{8}\]  
The replicator dynamic equation for retailers is:  
\[\frac{dy}{dt}=y(U_{RH}-\overline{U_{R}}) \tag{9}\]  
Substituting Equations (6)-(8) into Equation (9) and combining Equation (5) yield the following dynamic replicator system of long-term emission reduction game:  
\[\left\{ \begin{array}{l}dx/dt=x(1-x)[u_{M}(q_{1}-q_{2}-q_{3})y+u_{M}q_{2}-c_ {M}]\\ dy/dt=y(1-y)[u_{R}(q_{1}-q_{2}-q_{3})x+u_{R}q_{3}-c_{R}]\end{array} \right. \tag{10}\]  
**Proposition 1**_The above dynamic replicator system has four unconditional equilibrium points, i.e., \(E_{1}(0,0)\), \(E_{2}(1,0)\), \(E_{3}(0,1)\) and \(E_{4}(1,1)\). Furthermore, \(E_{5}(x_{0},y_{0})\) can be also an equilibrium point if (i) \(c_{M}<u_{M}q_{2}<u_{M}q_{1}<c_{M}+u_{M}q_{3}\), \(c_{R}<u_{R}q_{3}<u_{R}q_{1}<c_{R}+u_{R}q_{2}\), or (ii) \(u_{M}q_{2}<c_{M}<c_{M}+u_{M}q_{3}<u_{M}q_{1}\), \(u_{R}q_{3}<c_{R}<c_{R}+u_{R}q_{2}<u_{R}q_{1}\), \(0<x_{0}<1\), \(0<y_{0}<1\) where \(x_{0}=(c_{R}-u_{R}q_{3})/\left[u_{R}(q_{1}-q_{2}-q_{3})\right]\), \(y_{0}=(c_{M}-u_{M}q_{2})/\left[u_{M}(q_{1}-q_{2}-q_{3})\right]\). However, the equilibrium \(E_{5}(x_{0},y_{0})\) does not exist if \(q_{1}=q_{2}+q_{3}\). The proof is shown in the Appendix A.