# Evolution Game Analysis of Supply Chain Synergy Benefits of Prefabricated Building Projects
### Game Model Construction  
According to evolutionary game theory, the game will automatically evolve towards the optimal strategy of the players [37]. The participants of the game have two different probabilistic strategies, which form a set of strategies. In the prefabricated building project supply chain collaboration process, the government can choose whether or not to provide policy subsidies for other participating subjects in the supply chain and whether or not to supervise and regulate the collaboration process between the subjects. The government can choose from the set of strategies (subsidy, no subsidy), which consists of the probability of government subsidy for (x) and the probability of government subsidy for (\(1-x\)). The set of strategies that the construction unit can choose is (synergy, no synergy), where the probability that the construction unit chooses to participate in the supply chain synergy of the prefabricated building project is (y), and the probability that the construction unit  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Symbolic Description** \\ \hline G & Governments \\ P & The Construction Unit \\ F & Prefabricated Component Manufactures \\ t\({}_{1}\) & Costs to the government for providing subsidies and monitoring supply chain operations \\ c & Synergy costs arising from cooperation between construction units and precast manufacturers \\ m & Government subsidies to construction units and precast manufacturers \\ k & Cost-sharing ratio factor between the construction unit and the fabricator \\ P1 & Benefits to the government when it chooses the “subsidy” strategy \\ b & The proportion of government benefits from choosing the “no-subsidy” strategy as a percentage of benefits from choosing the “subsidy” strategy \\ p2 & Initial benefits before the construction unit participates in supply chain synergies \\ p3 & Initial benefits for precast producers before they participate in supply chain synergies \\ p & Supply chain synergy incentive benefits \\ a & Incentive benefit sharing factor for construction units \\ (\(1-a\)) & Incentive benefit sharing factor for forecasters \\ d\({}_{1}\) & Benefits to precast producers when they choose to cooperate and to construction units when they choose to produce independently \\ d\({}_{2}\) & Proceeds when the construction unit chooses to cooperate and the precast manufacturer chooses to produce independently \\ t\({}_{2}\) & Government financial subsidies to precast component dealers \\ l\({}_{1}\) & Compensation is to be paid to the construction unit in case of default by the precast manufacturer \\ l\({}_{2}\) & Compensation to be paid to precast manufacturers in the event of default by the construction unit \\ \hline \hline \end{tabular}
\end{table}
Table 1: The meaning of model parameter symbols.  
chooses not to participate in the synergy is (1 - y). The set of strategies available to prefabricated component manufacturers is (synergistic, non-synergistic), the probability that a prefabricated component manufacturer chooses to participate in the supply chain synergy of a prefabricated building project is (z), and the probability that the manufacturer chooses not to participate in the synergy is (1 - z).  
According to the model assumptions, the game payment matrix of the three parties involved in the supply chain collaboration of prefabricated building projects is constructed, as shown in Table 2. Where the cells from (three rows and three columns) to (six rows and four columns) denote the return for each participating subject under that strategy choice. The first row of these cells, Ai (i = 1, 2, \(\ldots\), 8) denotes the revenue function of the government, and the second row Bi (i = 1, 2, \(\ldots\), 8) denotes the revenue function of the construction unit, and the third row Ci (i = 1, 2, \(\ldots\), 8) denotes the revenue function of the precast manufacturer.