## 3 Evolutionary Game Model for Carbon Emission Reduction
### Managerial Insights
That is, these elaborated conditions: (1) Comparing with no cooperative emission reduction, retailer's unilateral participation in emission reduction may increase her profit, whereas manufacturer's unilateral involvement will decrease his profit; moreover, when comparing with the profits under cooperative emission reduction, retailer's free riding action decreases her own profit but manufacturer's free riding action increase manufacturer's profit ((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) Unilateral emission reduction for manufacturers and retailers is anyhow better than no emission reduction; furthermore, retailer's profit under bilateral cooperative emission reduction is more than that with his/her free-riding behaviors, however, manufacturer's profit under bilateral cooperative emission reduction less than that with his/her free-riding actions ((ii) \(c_{M}<u_{M}q_{2}<u_{M}q_{1}<u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}>c_{R}\), \(u_{R}q_{1}>u_{R}q_{2}+c_{R}\)); (3) For the retailer, unilateral involvement can generate more profit than no participation in emission reduction, whereas for manufacturer his unilateral emission reducing incurs less profit than no emission; moreover, free-riding again cause more profit than bilateral emission reduction ((iii) \(u_{M}q_{2}<c_{M}\), \(u_{M}q_{1}<u_{M}q_{3}+c_{M}\) and \(c_{R}<u_{R}q_{3}<u_{R}q_{1}<u_{R}q_{2}+c_{R}\)). The above explanation shows that for manufacturers increasing input on emission reduction might have a weak effect on profit and emission due to possible emission-reducing technology bottleneck. However, for retailers grasping the increase of consumers awareness to low carbon can substantially impact the system performance. **Observation 4** Manufacturers' and retailers' actions will converge to \(E_{4}(1,1)\) with evolutionary stable equilibrium strategy \((M_{H},R_{H})\), i.e., they both choose high-level emissions-reduction/promotion strategy when one of three conditions as follows holds, namely, (i) \(u_{M}q_{2}<c_{M}<u_{M}q_{3}+c_{M}<u_{M}q_{1}\) and \(u_{R}q_{3}>c_{R}\), \(u_{R}q_{1}>u_{R}q_{2}+c_{R}\) or (ii) \(u_{M}q_{2}>c_{M}\), \(u_{M}q_{1}>u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}<c_{R}<u_{R}q_{2}+c_{R}<u_{R}q_{1}\), or (iii) \(u_{M}q_{2}>c_{M}\), \(u_{M}q_{1}>u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}>c_{R}\), \(u_{R}q_{1}>u_{R}q_{2}+c_{R}\), that is, elaborately, (1) Comparing with no emission reduction, unilateral participation in emission reduction is beneficial to the retailer, yet adverse to manufacturer; moreover, for manufacturers and retailers bilateral cooperation on emission reduction can generate more profit than that with free riding behavior ((i) \(u_{M}q_{2}<c_{M}<u_{M}q_{3}+c_{M}<u_{M}q_{1}\) and \(u_{R}q_{3}>c_{R}\), \(u_{R}q_{1}>u_{R}q_{2}+c_{R}\)); or (2) Comparing with no emission reduction, unilateral participation in emission reduction is beneficial to manufacturers, yet adverse to retailers; furthermore, for manufacturers and retailers bilateral cooperation on emission reduction will generate more profit than that with free riding behavior ((ii) \(u_{M}q_{2}>c_{M}\), \(u_{M}q_{1}>u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}>c_{R}<u_{R}q_{2}+c_{R}<u_{R}q_{1}\)); or (3) Comparing with no emission reduction, unilateral participation in emission reduction is beneficial to retailers and manufacturers; what is more, bilateral cooperation on emission reduction will bring about more profit than that with free riding actions ((iii) \(u_{M}q_{2}>c_{M}\), \(u_{M}q_{1}>u_{M}q_{3}+c_{M}\) and \(u_{R}q_{3}>c_{R}\), \(u_{R}q_{1}>u_{R}q_{2}+c_{R}\)). The above Observation implies some insights as follows. For either the manufacturer or the retailer, the profit increment by bilateral cooperative emission reduction is greater than that with his or her free-riding behavior, respectively. Therefore, both sides have the incentive to participate in emission reduction. Note that two special cases in Proposition 2 are stated in Observation 5 and 6 as follows.