## 3 Model and Analysis
### Construction and Analysis of Stochastic Evolutionary Game Model
#### 3.3.2 Game Model Analysis
The data for Conditions 1 and 2 are presented in \[LV(t,x)=[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y} +(1-z)N_{Z}+(1-u)N_{U}+G_{X}]x \tag{13}\]3 Based on these datasets, 30 repeated simulations were conducted for each condition, with the results depicted in \[LV(t,x)=[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y} +(1-z)N_{Z}+(1-u)N_{U}+G_{X}]x \tag{13}\]4 The stable states achieved in each simulation aligned with the analytical outcomes. On average, Conditions 1 and 2 required approximately 163 and 40 iterations, respectively, to reach stability. Due to random disturbance factors, fluctuations occurred after system stabilization, which were more pronounced in Condition 1, with a strategy probability fluctuation range of [0, 0.07]. In contrast, Condition 2 exhibited no fluctuations. This discrepancy arises because the parameters in Condition 2 were set in an overly ideal manner, rendering random disturbances insufficient to impact the strategy. Experiments validate the reliability of the stability analysis. With other parameters held constant and only the intensity of random interference varied, the system evolution is depicted in \[LV(t,x)=[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y} +(1-z)N_{Z}+(1-u)N_{U}+G_{X}]x \tag{13}\]5 The system's final stability remains unaffected. However, as the interference intensity increases, the fluctuation range of strategy probabilities in the stable state also widens. \[LV(t,x)=[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y} +(1-z)N_{Z}+(1-u)N_{U}+G_{X}]x \tag{13}\]6 System evolution paths under varying random noise intensity. (**a**) Condition 1; (**b**) Condition 2. \[LV(t,x)=[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y} +(1-z)N_{Z}+(1-u)N_{U}+G_{X}]x \tag{13}\]7 System evolution paths corresponding to theoretical analysis (30 simulations). (**a**) Condition 1; (**b**) Condition 2. \begin{table}
\begin{tabular}{c l} \hline \hline
**Conditions** & **Numerical Setting** & **Stability Point** \\ \hline \multirow{4}{*}{Condition 1: \(\{(X1\cup X2)\}\)} & \(x=y=z=u=0.2,\sigma=0.05,E_{X}=7,E_{Y}=10,E_{Z}=E_{U}=4\), \\  & \(G_{X}=G_{Y}=G_{Z}=G_{U}=2,C_{X}=C_{Y}=18,C_{Z}=C_{U}=13\), \\  & \(N_{Y}=N_{Z}=N_{U}=1;I=1,\alpha C=1,D=5\), \\  & \(\gamma_{I}=0.5,A_{i}=0.2,\beta_{i}=0.4,(i+j),i=(X,Y,Z,U)\); \\  & \(a_{ij}=0.3(i=\{Y,Z,U\};j=\{X,Y,Z\};i\
eq j)\) \\ \hline \multirow{4}{*}{Condition 2: \(\{(X3\cup X4)\}\)} & \(x=y=z=u=0.7,\sigma=0.05,E_{X}=7,E_{Y}=9,E_{Z}=E_{U}=5\), \\  & \(G_{X}=G_{Y}=G_{Z}=G_{U}=3,C_{X}=C_{Y}=21,C_{Z}=C_{U}=13\), \\  & \(N_{Y}=N_{Z}=N_{U}=1.5;I=1,\alpha C=1,D=5\) \\  & \(\gamma_{I}=0.5,A_{i}=0.8,\beta_{i}=0.5,(i\
eq j),i=(X,Y,Z,U\};\) \\  & \(a_{ij}=0.3(i=\{Y,Z,U\};j=\{X,Y,Z\};i\
eq j)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameter settings for stability analysis of game systems. Random disturbance variables do not affect whether the system reaches a stabilization point. Unlike the traditional deterministic evolutionary game model [41], this model's evolution to a stable point depends on agents' influencing factors and strategic choices. This aligns with the actual scenario where businesses collaborate on PBSC carbon emission reduction based on observing each other's choices.