## 3 Evolutionary Game Model for Carbon Emission Reduction
### Managerial Insights  
In this subsection, analyzing the evolutionary stable strategies in proposition 2 summarizes the managerial implications as below. **Observation 1** For manufacturers and retailers, the system actions will converge to \(E_{1}(0,0)\) with evolutionary stable equilibrium strategy \((M_{L},R_{L})\), that is, manufacturers and retailers both choose low-level emission reduction and promotion strategies, respectively, when parameters satisfy conditions (i) \(u_{M}q_{2}<c_{M}\), \(u_{M}q_{1}<u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}<c_{R}\), \(u_{R}q_{1}<u_{R}q_{2}+c_{R}\), or (ii) \(u_{M}q_{2}<c_{M}<u_{M}q_{3}+c_{M}<u_{M}q_{1}\) and \(u_{R}q_{3}<c_{R}\), \(u_{R}q_{1}<u_{R}q_{2}+c_{R}\), or (iii) \(u_{M}q_{2}<c_{M}\), \(u_{M}q_{1}<u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}<c_{R}<u_{R}q_{2}+c_{R}<u_{R}q_{1}\). That is, for either the manufacturer or the retailer, the firm's profit from unilaterally participating in emission reduction is less than that under no participation in emission reduction, if one of the three conditions below holds: (1) For the manufacturer and retailer, their total profit under emission-reduction cooperation between two parties is less than that in the setting where there exists the free- riding behavior ((i) \(u_{M}q_{2}<c_{M}\), \(u_{M}q_{1}<u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}<c_{R}\), \(u_{R}q_{1}<u_{R}q_{2}+c_{R}\)); (2) Comparing with the cooperative emission reduction, manufacturer's free-riding behavior will decrease his own individual profit, while retailer's free-riding behavior will increase her own profit ((ii) \(u_{M}q_{2}<c_{M}<u_{M}q_{3}+c_{M}<u_{M}q_{1}\) and \(u_{R}q_{3}<c_{R}\), \(u_{R}q_{1}<u_{R}q_{2}+c_{R}\)); and (3) Still referring to the cooperative emission reduction, manufacturer's free-riding behavior will increase his own profit, while retailer's free-riding behavior will decrease her own profit ((iii) \(u_{M}q_{2}<c_{M}\), \(u_{M}q_{1}<u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}<c_{R}<u_{R}q_{2}+c_{R}<u_{R}q_{1}\)), the evolutionary stable equilibrium strategies for manufacturers and retailers is \((M_{L},R_{L})\). This Observation has some implications to practice. Retailers' high-level low-carbon promotion has limited effect on the demand (i.e., \(q_{3}\) is small or \(c_{R}\) is large) and in the case that consumers have a full understanding of low carbon products; Meanwhile, manufacturers may confront bottleneck in reducing emission when high-level input on emission reduction contributes little to revenue increase(i.e., \(q_{2}\) is small or \(c_{M}\) large). Thus, profit maximizing will be the dominant principle for decision making resulting in those two parties both adopting low-level emission reduction strategy become the ESS, which usually takes place at the late period of the life cycle of low-carbon products. **Observation 2** For manufacturers and retailers, the system actions will converge to \(E_{2}(1,0)\) with evolutionary stable equilibrium strategy \((M_{H},R_{L})\), i.e., manufacturers choose high-level emissions reduction strategy and retailers choose low-level promotion strategy, respectively; when parameters satisfy (i) \(c_{M}<u_{M}q_{2}\), \(u_{M}q_{1}>u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}<c_{R}\), \(u_{R}q_{1}<u_{R}q_{2}+c_{R}\), or (ii) \(c_{M}<u_{M}q_{2}\), \(u_{M}q_{1}>u_{M}q_{3}+c_{M}\) and \(c_{R}<u_{R}q_{3}<u_{R}q_{1}<u_{R}q_{2}+c_{R}\), or (iii) \(c_{M}<u_{M}q_{2}<u_{M}q_{1}<u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}<c_{R}\), \(u_{R}q_{1}<u_{R}q_{2}+c_{R}\).