## 3 Evolutionary Game Model for Carbon Emission Reduction
### Managerial Insights
**Observation 5** When parameters satisfy \(u_{M}q_{2}<c_{M}<u_{M}q_{3}+c_{M}<u_{M}q_{1}\) and \(u_{R}q_{3}<c_{R}<\)\(u_{R}q_{2}+c_{R}<u_{R}q_{1}\), in other words, for manufacturers and retailers, unilateral participation in emission reduction will incur less profit than that by no emission reduction, while bilateral cooperative emission reduction may bring about more profit than free riding behavior; in addition, the demand increment by cooperative emission reduction is more than separate emission reduction (\(q_{1}>q_{2}+q_{3}\)), the system might converge to \(E_{1}(0,0)\) or \(E_{4}(1,1)\). Finally, the system converges to which stable point determined by the initial probability of two parties' choice for emission reduction strategy \((x,y)\). We show the phase diagram under the condition in Figure 1 with two unstable point,\(E_{2}(1,0)\) and \(E_{3}(0,1)\), a saddle point \(E_{5}(x_{0},y_{0})\) which converges to the boundary of two stable points \(E_{1}(0,0)\) and \(E_{4}(1,1)\). If initial probability is at below the polyline \(A\) of area, the system finally evolves to the steady point \(E_{1}(0,0)\), where both sides don't participate in emission reduction. On the contrary, in case that the initial probability is over the polyline of \(B\) region, the system eventually evolves to a stable point \(E_{4}(1,1)\), where both parties participate in emission reduction. **Observation 6** When parameters satisfy \(c_{M}<u_{M}q_{2}<u_{M}q_{1}<u_{M}q_{3}+c_{M}\) and \(c_{R}<u_{R}q_{3}<\)\(u_{R}q_{1}<u_{R}q_{2}+c_{R}\), namely, for manufacturers and retailers unilateral participation in emission reduction can entail more profit than that under no emission reduction, yet bilateral cooperative emission reduction unexpectedly incur less profit than that with free-riding behaviors; in addition, bilateral cooperative emission reduction will incur less total demand increment than unilateral emission reduction (\(q_{1}<q_{2}+q_{3}\)), the system maybe converge to \(E_{2}(1,0)\) or \(E_{3}(0,1)\). Similarly, the system converges to which stable point, which is determined by the initial probability of two parties' choice for emission reduction strategy \((x,y)\). The phase diagram under the condition can be shown as Observation (6) in Figure 1, where there are two unstable point, \(E_{1}(0,0)\) and \(E_{4}(1,1)\), a saddle point \(E_{5}(x_{0},y_{0})\) which converges to the boundary of two stable points, \(E_{2}(1,0)\) and \(E_{3}(0,1)\). If initial  
Figure 1: The dynamic phase diagram of system evolution under conclusion 5 and conclusion 6.  
probability is over the polyline of \(C\) area, the system finally evolves to the steady point \(E_{3}(0,1)\), where manufacturers adopt low level emission reduction strategy, however retailers choose high level emission reduction. On the contrary, if the initial probability is at below the polyline of \(D\) region, the system eventually evolves to a stable point \(E_{2}(1,0)\), where manufacturers adopt high level emission reduction strategy, however retailers choose low level emission reduction. In this study, we focus on the selection of long-term emission reduction cooperation strategies between manufacturers and retailers (manufacturer's high/low level emission reduction strategy and retailer's high/low level low-carbon promotion strategy), and whether the combination of these strategies is stable, and further discuss which parameters affect these stability strategies. Proposition 1 gives the equilibrium point of the choice of the two strategies. Proposition 2 discusses the stability of these equilibrium strategies in what parameter range. Conclusion 1-6 explains the management meaning of the parameter range of these evolutionary stability strategies, respectively, and gives the managerial insights and suggestions of long-term emission-reduction cooperation between manufacturers and retailers.