# Evolution Game Analysis of Supply Chain Synergy Benefits of Prefabricated Building Projects
## 3 The Prefabricated Building Supply Chain Synergistic Game Analysis
### Equilibrium Stability Analysis  
According to evolutionary game theory, the condition for satisfying the model's ESS is that all eigenvalues of the Jacobi matrix at that point are nonpositive. To analyze the sign of the corresponding eigenvalues of different ESSs, it may be assumed that the gains of the three subjects of the game when they do not engage in supply chain collaboration are smaller than the gains when they all participate in supply chain collaboration, \(-ap-l_{2}+d_{1}+ck-km<0\), \(-[(1-a)p+l_{1}-d_{2}+t_{2}-(1-k)(c-s)]<0\), \(t_{1}+t_{2}-(1-b)p_{1}<0\). The stability of the equilibrium point is discussed below in three scenarios:  
(1) Scenario 1: When \((1-k)(c-m)>l_{2}+t_{2}\) and \(k(c-m)>l_{1}\), the cost of collaboration invested by the precast manufacturer to participate in supply chain collaboration when the government subsidizes it is higher than the sum of the government's financial support to the precast manufacturer and the compensation paid by the construction unit to the precast manufacturer when the construction unit does not participate in supply chain collaboration. Moreover, the cost of collaboration invested by the construction unit when the government subsidizes it is higher than the compensation paid by the precast manufacturer to the construction unit when the precast manufacturer does not participate in collaboration. In short, both the builder and the precast manufacturer have invested more in costs than in benefits, respectively. In this scenario, according to Table 4, the ESSs are M\({}_{6}\) (1,0,0) and M\({}_{8}\) (1,1,1), which correspond to negative eigenvalues (\(-\)), and the corresponding evolutionary strategies of the participating entities are (subsidize, no synergy, no synergy) and (subsidize, synergy, synergy). (2) Scenario 2: When \(l_{1}>kc\) or \(l_{2}>(1-k)c\), the compensation required to be paid to the construction unit by the precast manufacturer for not participating in supply chain synergy is higher than the cost of synergy required to be invested by the construction unit when the government does not subsidize it, or the compensation required to be paid to the precast manufacturer by the construction unit for not participating in supply chain synergy is higher than the cost of synergy required to be invested by the precast manufacturer when the government does not subsidize it. In this scenario, according to Table 4, the ESS is M\({}_{8}\) (1,1,1), whose corresponding eigenvalue is negative (\(-\)), and the corresponding evolution strategy of the participating entities is (subsidy, synergy, synergy). \begin{table}
\begin{tabular}{c c c c} \hline
**ESS** & **The Eigenvalue \(\mathbf{\lambda_{1}}\)** & **The Eigenvalue \(\mathbf{\lambda_{2}}\)** & **The Eigenvalue \(\mathbf{\lambda_{3}}\)** \\ \hline M\({}_{1}\) (0, 0, 0) & \(l_{1}-kc\) & \((k-1)c+l_{2}\) & \((1-b)p_{1}-t_{1}\) \\ M\({}_{2}\) (0, 0, 1) & \(ap+l_{2}-d_{1}-ck\) & \((1-k)c-l_{2}\) & \((1-b)p_{1}-t_{1}-t_{2}\) \\ M\({}_{3}\) (0, 1, 1) & \(+d_{1}+ck-ap-l_{2}\) & \(-[(1-a)p-(1-k)c+l_{1}-d_{2}]\) & \((1-b)p_{1}-t_{1}-t_{2}\) \\ M\({}_{4}\) (0, 1, 0) & \(ck-l_{1}\) & \((1-a)p-(1-k)c+l_{1}-d_{2}\) & \((1-b)p_{1}-t_{1}\) \\ M\({}_{5}\) (1, 1, 0) & \(k(c-m)-l_{1}\) & \((1-a)p+l_{1}-d_{2}\) & \(t_{1}+t_{2}-(1-b)p_{1}\) \\  & \(+t_{2}-(1-k)(c-s)\) & \(t_{1}+t_{2}-(1-b)p_{1}\) \\ M\({}_{6}\) (1, 0, 0) & \(l_{1}-k(c+m)\) & \(-[t_{2}-(1-k)(c-m)+l_{2}]\) & \(t_{1}+t_{2}-(1-b)p_{1}\) \\ M\({}_{7}\) (1, 0, 1) & \(ap+l_{2}-d_{1}-ck+km\) & \(t_{2}-(1-k)(c-m)+l_{2}\) & \(t_{1}+t_{2}-(1-b)p_{1}\) \\ M\({}_{8}\) (1, 1, 1) & \(-ap-l_{2}+d_{1}+ck-km\) & \(-[(1-a)p+l_{1}-d_{2}\) & \(t_{1}+t_{2}-(1-b)p_{1}\) \\  & & \(+t_{2}-(1-k)(c-s)]\) & \\ \hline \end{tabular}
\end{table} 
Table 3: Eigenvalues of the Jacobi matrix.