## 5 Research on Evolutionary Game
### Duplicative Dynamic  
Suppose \(U_{2}(s_{1},s_{2})=U_{1}(s_{2},s_{1})=g\), \(U_{2}(s_{2},s_{1})=U_{1}(s_{1},s_{2})=h\), \(U_{2}(s_{1},s_{1})=U_{1}(s_{1},s_{1})=e\), and \(U_{2}(s_{2},s_{2})=U_{1}(s_{2},s_{2})=f\). The proportion of group members adopting the \(S_{1}\) strategy is \(x\), and the proportion adopting the \(S_{2}\) strategy is \(1-x\). \(E(U_{s_{1}})\), \(E(U_{s_{2}})\), and \(E(U)\) are used to represent the expected profit of the \(S_{1}\) strategy, the expected profit of the \(S_{2}\) strategy, and the average expected profit, respectively, as shown in Equations (12)-(14).  
\[E(U_{s_{1}})\ =x\cdot e+(1-x)\cdot h \tag{12}\]  
\[E(U_{s_{2}})=x\cdot g+(1-x)\cdot f \tag{13}\]  
\[E(U)=x\cdot E(U_{s_{1}})+\ (1-x)\cdot E(U_{s_{2}}) \tag{14}\]  
The evolutionary game will eventually reach a stable state, and the process of reaching a stable state is reflected by the duplicative dynamic equation. The duplicative dynamic is used to describe the dynamic strategy adjustment process of the players--the core of which is that the individuals who adopt the more successful strategies in the group will gradually increase. Duplicative dynamic equations are often represented by dynamic differential equations. Therefore, the duplicative dynamic equation of the above evolutionary game model can be expressed as Equation (15).  
\[\begin{split}\frac{\mathrm{d}x}{\mathrm{d}t}&=x \cdot[E(U_{s_{1}})-E(U)]=x\cdot[E(U_{s_{1}})-x\cdot E(U_{s_{1}})-(1-x)E(U_{s_ {2}})]\\ &=x\cdot(1-x)\cdot[x\cdot(e-g)+(1-x)\cdot(h-f)]\end{split} \tag{15}\]  
Bringing the expressions of \(g\), \(h\), \(e\), and \(f\) into Equation (15), the final duplicative dynamic equation of the evolutionary game is obtained as shown in Equation (16).  
\[\frac{\mathrm{d}x}{\mathrm{d}t}=x^{2}\cdot(1-x)\cdot(\frac{E_{k_{1}}}{q^{ \mathrm{on}}_{k_{1},i-1}+q^{\mathrm{wait}}_{i}}-\frac{E_{k_{2}}}{q^{\mathrm{ on}}_{k_{2},i-1}+\frac{1}{2}\cdot q^{\mathrm{wait}}_{i}})+x\cdot(1-x)^{2} \cdot(\frac{E_{k_{1}}}{q^{\mathrm{on}}_{k_{1},i-1}+\frac{1}{2}\cdot q^{ \mathrm{wait}}_{i}}-\frac{E_{k_{2}}}{q^{\mathrm{on}}_{k_{2},i-1}+q^{\mathrm{ wait}}_{i}}) \tag{16}\]  
The equation reflects the time evolution process of the proportion of passenger group members adopting the induced strategy, revealing the dynamic adjustment process of group behavior.  
Let \(\frac{\mathrm{d}x}{\mathrm{d}t}=0\), and the three stable states \(x^{*}_{1}\), \(x^{*}_{2}\), and \(x^{*}_{3}\) of the duplicative dynamic are solved by Equations (17)-(19)  
\[x^{*}_{1}=0 \tag{17}\]  
\[x^{*}_{2}=1 \tag{18}\]  
\[x^{*}_{3}=\frac{\frac{E_{k_{2}}}{q^{\mathrm{on}}_{k_{2},i-1}+q^{\mathrm{wait}}_ {i}}-\frac{2E_{k_{1}}}{2q^{\mathrm{on}}_{k_{1},i-1}+q^{\mathrm{wait}}_{i}}}{ \frac{E_{k_{1}}}{q^{\mathrm{on}}_{k_{1},i-1}+q^{\mathrm{wait}}_{i}}-\frac{2E_ {k_{2}}}{2q^{\mathrm{on}}_{k_{2},i-1}+q^{\mathrm{wait}}_{i}}+\frac{E_{k_{2}}}{ q^{\mathrm{on}}_{k_{2},i-1}+q^{\mathrm{wait}}_{i}}-\frac{2E_{k_{1}}}{2q^{\mathrm{on}}_{k_{1},i-1}+q^{ \mathrm{wait}}_{i}}} \tag{19}\]  
The steady state of the duplicative dynamic suggests that the proportion of the passenger population receiving the induction strategy may stabilize at \(x^{*}_{1}\), \(x^{*}_{2}\), or \(x^{*}_{3}\).