## 1 Introduction
For instance, Haier, one of Chinese main household appliances manufacturers, allies manufacturer GE, giant retailers, like Gome and Suning, Chinese Association of Environmental Protection Industry and some other organizations to form the first \"Zero-Carbon-Conversion Alliance\" in China. However, the free-rider problem, a well-known economics phenomenon, can still take place naturally in the aforementioned alliance with carbon-reduction cooperation. For example, when manufacturers strive to decrease emission intensity to achieve lower carbon in products attracting increased demand from consumers with low-carbon preference, from which the retailer can benefit even if she has done nothing. However, the free-riding phenomenon harms fairness, ruins the supply chain performance, and hampers the construction of long-run advantage for the entire supply chain. Motivated by this, one of our main objectives is to explore whether free riding behavior has impacts on the cooperation strategy of emission reduction among vertical supply chain cooperation in the long term. We therefore attempt to conduct evolutionary game theoretic analysis for the supply chain while considering the behaviors of emission reduction, low-carbon promotion, and free riding in the long-term cooperation. Moreover, we will investigate the impact of free riding behavior on the cooperative strategy of emission reduction in long-run vertical channel interaction. Although evolutionary game is widely used in the field of biology and economics, few researches considered the free riding problem within the vertical cooperation of carbon emission reduction. Hence, taking aim at this gap, we strive to focus on three research questions, as follows: (1) What is the evolutionary trend of carbon emission reduction under long-term cooperation in supply chains? (2) What affects evolutionary stable strategy of carbon emission reduction in a long-term cooperative supply chains? (3) What is the impact of the free-riding behavior on vertical cooperative strategy of emission reduction for supply chain enterprises in the long term? In response to these main questions, we construct evolutionary game models to depict chain members' behavioral patterns upon their interacting strategies. Then, we attain evolutionary stable strategies and their associated existing preconditions under various scenarios, which provides a lot of managerial insights, and particularly shows that the evolutionary stable strategy (ESS) of the channel carbon reduction and promotion is apparently affected by incremental profits attributed to factors, including each player's unilateral participation in emission reduction. Moreover, we explore a player's free-rider opportunistic practice in cooperative carbon reduction and joint participation in emission reduction. However, the profit increment degree will heavily affect the result of ESS. The remainder of this paper is organized as follows. In Section 2, we give a detailed review on related literature. In Section 3, we then address evolutionary game models for vertical cooperation of carbon emission reduction. In Section 4, we provide typical computational experiments and analysis. Concluding remarks and future research directions come in Section 5.