## 2 Literature Review  
In this section, we give a review on three streams of literature that are closely related to our research, i.e., vertical channel cooperation, low-carbon operations and the applications of evolutionary game theory in supply chain. In the field of supply chain management, many scholars report that vertical cooperation in supply chain can improve the chain performance [4; 5; 6]. A lot of literature in this kind mainly includes vertical cooperative advertising and research and development (R&D) and so on. The literature on vertical cooperative advertising mainly states that upstream manufacturers through sharing downstream advertising cost to stimulate retailers to enhance promotion ability, which thus generates more market demand, like [7], as the first study on this topic. Decades hereafter, the development of vertical cooperative advertising can be divided into short-term static cooperation and long-term dynamic cooperation, where the former kind mainly uses Stackelberg game to develop contract design of cooperative advertising in supply chain [8; 9], while our research belongs to the latter one. Some other researchers considered risk preference and vertical cooperation both for supply chain firms, like [10], and differential games for long-run cooperation (e.g., [11; 12; 13]). In this study, we simultaneously consider the behavior of retailers' low-carbon promotion and that of manufacturers' emission-reduction investment, while we primarily analyze the ESS of vertical cooperative behaviors on emission reduction upon the evolutionary game model. This is distinct from [PERSON] et al. [12] using differential game to get optimal feedback strategy. There are also a number of literature referring to vertical cooperative R&D, such as [14; 15; 16; 17; 18]. Remarkably, some literature studies the contract design for solving free-riding problem emerging from R&D technology spillover in the supply chain, like [19; 20]. However, different from previous literature, we focus on cooperative carbon emission reduction coupled with the low-carbon promotion, particularly from the viewpoint of evolutionary game theoretic analysis. Recently, the low-carbon supply chain management has been extensively studied, such as [21; 22; 23; 24; 25]. Our paper is related to literature involving consumers' sensitivity on low-carbon products. [26] consider consumers' low-carbon awareness based on traditional supply chain management. Assuming demand dependent on emission reduction level, [27] discussed the effects of several contracts on supply chain performances as well as the carbon reduction effectiveness over different contracts. The literature [28] studied cooperative emission reduction in retailer-dominant and power-balanced supply chains and further discussed coordinating the emission reduction activities based on cost sharing contract and wholesale price premium contract, respectively. Other more recent researches on low-carbon supply chain management include [29; 30; 31]. Generally, the existing literature usually discusses single-cycle joint emission reduction problem, whereas the current paper explores vertical cooperation of long-term emission reduction and low-carbon promotion in supply chain from the evolutionary process angle. Our study is also related to the applications of evolutionary game theory in supply chain management. Evolutionary game theory is a combination of game theory and dynamic evolution process analysis, which emphasizes the analysis of dynamic equilibrium strategy and stable results. Evolutionary game model is not a monocyclic optimization model but it is concerned with how the players choose strategies when the time is approaching infinity, and furthermore discuss the stability of this strategy selection behavior.