## 3 Evolutionary Game Model for Carbon Emission Reduction  
### Preliminaries and Notations  
We consider a group of manufacturers and also a group of retailers in two-echelon supply chains, both of whom are homogeneous and profit-maximizing in their own group. Suppose that each manufacturer sells products through multiple retailers, and each retailer will purchase products from multiple manufacturers. Every manufacturer needs to pay a certain cost due to the governmental constraints for energy consumption per unit product, even if he quits the cooperation of reducing emissions. Assuming that consumers have the willingness to pay for environment-friendly products. Therefore, manufacturers have strategies of high-level emission reduction and low-level emission reduction while considering the cost input and consumers preference. For retailers, the promotion of low-carbon products can elevate consumers' low-carbon awareness resulting in the increase of market demand. Accordingly, retailers also hold strategies of high-level promotion and low-level promotion due to the similar reason to manufacturers. Supposing manufacturers and retailers have bounded rationality, hence it is hard for one party to accurately know the other's profit function and strategic choice. In other words, it is difficult to keep the optimal strategies the same over time. All of the players will adjust their choices by learning continually. In long-term cooperation on emission-reduction in supply chains, the population of manufacturers and retailers play multiple games randomly by continuous adjustment and strategic improvement, which eventually leads to the evolutionary equilibrium. Based on the above description, the evolutionary game model is used to analyze evolutionary stable strategies for two groups.  
The notations used in this paper are described, as follows.  
\begin{tabular}{l l} \hline \hline Notation/Description \\ \hline \(M_{H}\): & Manufacturers' strategy of high-level input on emission \\ \(M_{L}\): & Manufacturers' strategy of low-level input on emission \\ \(R_{H}\): & Retailers' strategy of high-level promotion on low carbon \\ \(R_{L}\): & Retailers' strategy of low-level promotion on low carbon \\ \(u_{M}\): & Constant marginal profit for each manufacturer, \(u_{M}>0\) \\ \(u_{R}\): & Constant marginal profit for each retailer, \(u_{R}>0\) \\ \(c_{M}\): & Difference of emission-reduction cost between high-level and low-level strategies for manufacturers \\ \(c_{R}\): & Difference of emission-reduction cost between high-level and low-level strategies for retailers \\ \(r_{M}\): & Net income of manufacturers when manufacturers and retailers both adopt low-level emission \\ \(r_{R}\): & Net income of retailers when manufacturers and retailers both adopt low-level emission reduction \\  & and promotion strategies, \(r_{R}>0\) \\ \(q_{1}\): & Sales increment when manufacturers and retailers adopt high-level emission reduction strategy \\ \(q_{2}\): & Sales increment when manufacturers adopt the strategy of high-level emission reduction, while \\ retailers use low-level low carbon promotion strategy (retailers' free riding behavior), \(q_{1}>q_{2}>0\) \\ \(q_{3}\): & Sales increment when manufacturers use the strategy of low-level emission reduction, while retailers \\  & adopt high-level low carbon promotion strategy (manufacturers' free riding), \(q_{1}>q_{3}>0\) \\ \(u_{M}q_{3}\): & The profit of manufacturers from free riding \\ \(u_{R}q_{2}\): & The profit of retailers from free riding \\ \hline \hline \end{tabular}