## 4 Results  
### Parameter Description  
Numerical simulations were performed using the software MATLAB R2023a. Referring to the literature [47; 48] and analysis, the initial strategy probabilities x, y, z, and u are set to 0.5; the time span is 400, the step size is 0.005, and \(\sigma=0.5\).  
For key factor parameter settings, relevant research findings and laws and regulations were referenced, and PBSC industry experts were consulted to ensure parameter rationality. This study surveyed PBSC practitioners and subject-matter experts via telephone or in-person interviews from universities, construction companies, and consulting firms in Beijing, Baoding, and Shenyang to ensure the statistics aligned with reality. They have offered insightful recommendations for this study, having participated in PB projects or researched PBSC and carbon emissions. The main topics included the input-output relationship of businesses in reducing carbon emissions, the percentage of carbon emission reduction costs at various PBSC stages, and the impact of businesses' emission reduction on other businesses' revenue. Detailed interview information is provided in Appendix A. In addition, after the initial parameter settings based on different references, the parameters were reviewed by experts to assess data rationality. Fine-tuning was conducted according to expert feedback to ensure scientific validity. The parameter settings are based on the following references and expert opinion.  
* For PBs, a higher prefabrication ratio significantly reduces carbon emissions [52; 72]. Based on the indicators for estimating investment in assembly building projects [73], which provide cost indicators for various types of assembly buildings (PBs), and considering that the cost increase for constructing buildings with a high assembly rate compared to those with a low assembly rate ranges from approximately 12% to 30%, it is abstractly assumed that this incremental cost represents the additional cost incurred by participants in implementing the abatement strategy.
* The regulation on energy conservation in civil buildings stipulates that the government imposes fines of 2-4% of the contract price on companies that exceed emissions [74].  
* The Civil Code states that the penalty for breach of contract may not be higher than 30% of the cost of damages [75].
* The Annual Report on China's carbon market indicates that the carbon trading market offers about a 50% reduction or exemption on carbon trading fees for emission-reducing entities [76]. Several places have also innovated the forms of carbon trading and carbon financial mechanisms, aiming to reduce the cost of emission reduction by lowering the transaction costs of emission-control enterprises [76].
* China grades PBs according to the Evaluation Standard for Assembled Buildings [77], with the higher the grade, the better the environmental benefits [72]. In Beijing, Shanghai, Guangdong, and Shenzhen, financial subsidy policies for assembled buildings with a grade of AA or above provide bonuses of RMB 30-120 per square meter, or approximately 5-10% of the unit cost.
* Referring to the research results related to collaborative emission reduction [41; 58; 78; 79], \(\beta_{ij}\in[0.5,1.0],\alpha_{ij}\in[0.3,0.6],\alpha_{ij}\leq\beta_{ij}\), set \(\beta_{ij}=0.5\), \((i=\{X,Y,Z,U\};j=\{X,Y,Z,U\};i\
eq j);\alpha_{ij}=0.3(i=\{Y,Z,U\};j=\{X,Y,Z\};i \
eq j)\). Based on the research findings on the impact of information on decision-making [56; 80], and considering the specific circumstances and expert opinions regarding PBSC emission reduction information, we set \(D=5,\gamma_{i}=0.5,\lambda_{i}=0.8\); \((i=\{X,Y,Z,U\})\).  
In addition to the above reference basis, we refer to recent literature on PBSC carbon accounting [55; 81; 82] to ensure data reliability. The parameters are repeatedly tested and adjusted for optimal visualization and intuitive problem explanation. Table 6 shows the remaining parameters. All parameters reflect realistic relationships for theoretical analysis, not actual monetary values.