## 3 Model and Analysis
### Construction of a Four-Way Evolutionary Game Model  
#### 3.2.1 Construction of the Payoff Matrix  
Based on the basic assumptions and parameters in the previous section, the following assumptions are made to establish a four-way evolutionary game model:  
**H1. **: _When the agent implements the \"abatement\" strategy, the support it receives from the external environment is \(\Pi i=Ji+\Delta Ci+D\gamma i\lambda i\). When the agent reduces emissions, the government provides green subsidies, and industry bodies offer reputational incentives, which can enhance economic or reputational strength. When the agent reduces emissions, it inevitably generates financing, purchases of equipment, and trades in environmentally friendly materials. Financial institutions streamline processes to cut costs, research bodies reduce equipment costs via joint R&D, and suppliers offer long-term deals for mutual benefits. These measures reduce the abatement cost to the agent. When an agent reduces emissions, it must acquire information on technologies, prices, and policies. The agent's information acquisition is bounded by availability and gathering ability. Post-acquisition, the agent analyzes it to make decisions, with the benefit being \(D\gamma i\lambda i\)[56]. When the agent implements the \"abatement\" strategy, the cost it pays is \(C_{i}\), and the benefit it obtains is \(E_{i}\), where \(i\in\{X,Y,Z,U\}\). These are the direct costs and benefits associated with implementing emission reduction. Costs include using eco-friendly materials, acquiring raw resources, and introducing advanced technology. Benefits are direct gains from emission reductions, such as energy savings, efficiency improvements, and enhanced market popularity._  
**H2. **: _If any two agents in PBSC implement the \"abatement\" strategy, the value-added benefit to agent \(i\) who implements the \"abatement\" strategy is \(\beta_{ij}E_{i}\), where \(i\in\{X,Y,Z,U\}\), \(j\in\{X,Y,Z,U\},i\
eq j\)[41]. The value-added benefits accumulate if multiple agents implement the \"abatement\" strategy. When two agents reduce emissions simultaneously, they form an alliance, share information, and collaborate on construction plans. This optimizes PBSC operation efficiency, reduces communication costs, and yields joint emission reduction benefits._  
**H3. **: _When upstream party \(j\) implements an \"abatement\" strategy and downstream party \(i\) implements a \"no-abatement\" strategy, downstream party \(i\) receives a free-rider benefit of \(\alpha_{ij}E_{i}\), where \(i\in\{Y,Z,U\},j\in\{X,Y,Z\},i\
eq j\)[41]. For example, green design on the design side reduces PB energy consumption and operating costs, while green construction by the construction side eases demolition difficulties and indirectly boosts revenue. For downstream parties, free-rider benefits from different upstream parties can accumulate._  
**H4. **: _When a party implements a \"no-abatement\" strategy, it incurs a loss of \(G_{i}(i\in\{X,Y,Z,U\})\), e.g., a fine from the regulator due to actions taken by organizations external to the PBSC [28]. Specifically, contractors not implementing abatement strategies will pollute the environment during construction, endangering neighboring residents' health and leading to complaints. Excessive contractor contamination can lead to fines or corrective action deadlines from regulators, causing financial losses. High energy consumption during operation and maintenance can reduce consumer willingness to buy, affecting revenue._  
**H5. **: _When X implements an \"abatement\" strategy, the downstream party will be contractually obligated to pay a penalty of \(N_{i}(i\in\{Y,Z,U\})\) when it implements a \"no-abatement\" strategy. If the builder decides to reduce emissions, the contract will include relevant agreements, such as using environmentally friendly materials and specifying work quality requirements.