## 3 Model and Analysis
### Construction and Analysis of Stochastic Evolutionary Game Model
#### 3.3.2 Game Model Analysis
**: _It should satisfy \(u\leq[-1-(1+x\beta_{ZX}+y\beta_{ZY}-xa_{ZX}-ya_{ZY})E_{Z}+C_{Z}-\Pi_{Z}\)\(-G_{Z}-xN_{Z}]/\beta_{ZU}E_{Z}\) and \(0\leq-1-(1+x\beta_{ZX}+y\beta_{ZY}-xa_{ZX}-ya_{ZY})E_{Z}+C_{Z}-\Pi_{Z}-G_{Z}-xN_{Z}\)._  
**U1. **: _It should satisfy \(x\leq[-1-(1+y\beta_{UY}+z\beta_{UZ}-ya_{UY}-za_{UZ})E_{U}+C_{U}-\Pi_{U}-G_{U}]/ (\beta_{UX}E_{U}-a_{UX}E_{U}+N_{U})\) and \(0\leq-1-(1+y\beta_{UY}+z\beta_{UZ}-ya_{UY}-za_{UZ})E_{U}+C_{U}-\Pi_{U}-G_{U}\)._  
If the P-order moment exponents of the zero solutions of Equations (13)-(16) are unstable, then  
\[x[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y}+(1-z)N _{Z}+(1-u)N_{U}+G_{X}]\geq x \tag{21}\]  
\[LV(t,x)=[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y} +(1-z)N_{Z}+(1-u)N_{U}+G_{X}]x \tag{13}\]0  
\[LV(t,x)=[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y} +(1-z)N_{Z}+(1-u)N_{U}+G_{X}]x \tag{13}\]1  
\[LV(t,x)=[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y} +(1-z)N_{Z}+(1-u)N_{U}+G_{X}]x \tag{13}\]2  
The condition satisfying Equation (21) is derived as X3 or X4 by performing the reduction computation. **X3. **: _When \(x\in(0,1]\) and \(\beta_{XY}E_{X}-N_{Y}>0,y\geq[1-(1+z\beta_{XZ}+u\beta_{XU})E_{X}+C_{X}-\Pi_{X} -N_{Y}\)\(-(1-z)N_{Z}-(1-u)N_{U}-G_{X}]/(\beta_{XY}E_{X}-N_{Y})\) and \(1-(1+z\beta_{XZ}+u\beta_{XU})E_{X}+C_{X}-\Pi_{X}-N_{Y}-(1-z)N_{Z}-(1-u)N_{U}-G_ {X}\leq\beta_{XY}E_{X}-N_{Y}\) should be satisfied._  
**X4. **: _When \(x\in(0,1]\) and \(\beta_{XY}E_{X}-N_{Y}<0,y\leq[1-(1+z\beta_{XZ}+u\beta_{XU})E_{X}+C_{X}-\Pi_{X} -N_{Y}\)\(-(1-z)N_{Z}-(1-u)N_{U}-G_{X}]/(\beta_{XY}E_{X}-N_{Y})\) and \(1-(1+z\beta_{XZ}+u\beta_{XU})E_{X}+C_{X}-\Pi_{X}-N_{Y}-(1-z)N_{Z}-(1-u)N_{U}-G_ {X}\leq\beta_{XY}E_{X}-N_{Y}\) should be satisfied._  
Similarly, the conditions for satisfying Equations (22)-(24) are Y2, Z2, and U2, respectively. **Y2. **: _It should satisfy \(z\geq[1-(1+x\beta_{YX}+u\beta_{YU}-xa_{YX})E_{Y}+C_{Y}-\Pi_{Y}-G_{Y}-xN_{Y}]/ \beta_{YZ}E_{Y}\) and \(1-(1+x\beta_{YX}+u\beta_{YU}-xa_{YX})E_{Y}+C_{Y}-\Pi_{Y}-G_{Y}-xN_{Y}\leq\beta_ {YZ}E_{Y}\)._  
**Z2. **: _It should satisfy \(u\geq[1-(1+x\beta_{ZX}+y\beta_{ZY}-xa_{ZX}-ya_{ZY})E_{Z}+C_{Z}-\Pi_{Z}-G_{Z}- xN_{Z}]/\beta_{ZU}E_{Z}\) and \(1-(1+x\beta_{ZX}+y\beta_{ZY}-xa_{ZX}-ya_{ZY})E_{Z}+C_{Z}-\Pi_{Z}-G_{Z}-xN_{Z} \leq\beta_{ZU}E_{Z}\)._  
**U2. **: _It should satisfy \(x\geq[1-(1+y\beta_{UY}+z\beta_{UZ}-ya_{UY}-za_{UZ})E_{U}+C_{U}-\Pi_{U}-G_{U}]/( \beta_{UX}E_{U}-a_{UX}E_{U}+N_{U})\) and \(1-(1+y\beta_{UY}+z\beta_{UZ}-ya_{UY}-za_{UZ})E_{U}+C_{U}-\Pi_{U}-G_{U}\leq\beta_ {UX}E_{U}-a_{UX}E_{U}+N_{U}\)._  
When Condition 1 is satisfied, namely \(\{(X1\cup X2)\cap Y1\cap Z1\cap U1\}\), each agent chooses to implement the \"no-abatement\" strategy, and the game system reaches the state of \((0,0,0,0)\). At present, the proportion of emission-reducing enterprises in PBSC is low, and their willingness to reduce emissions is weak. For each entity, the benefits from implementing \"abatement\" are lower than those from \"no-abatement\". This is the most pessimistic state when all the agents in PBSC are in a wait-and-see mode. When Condition 2 is satisfied, namely \(\{(X3\cup X4)\cap Y2\cap Z2\cap U2\}\), each agent chooses to implement the \"abatement\" strategy, and the game system reaches the state of \((1,\,1,\,1,\,1)\). At present, the proportion of emission-reducing enterprises in PBSC is relatively high. For each agent, the benefits from implementing \"abatement\" exceed those from \"no-abatement\". This is the ideal state in which all the agents in PBSC choose to implement an \"abatement\" strategy.