# Exploring and Modeling the Incentive Strategies of New Energy Ship Application in Yangtze River
## 3 Model Analysis
### Analysis of the Evolutionary Stability Strategies of Shipping Enterprises  
The expected payoff of shipping enterprises choosing NESs is as follows:  
\[\mathrm{E_{s1}=y[B_{1}+g(m)s-C_{1}-T+T_{1}-U_{0}]+(1-y)(B_{1}-C_{1}-T-U_{0})} \tag{1}\]  
The expected payoff for shipping enterprises choosing TFSs is as follows:  
\[\mathrm{E_{s2}=y(B_{2}-C_{2}-T-T_{2})+(1-y)(B_{2}-C_{2}-T)} \tag{2}\] \begin{table}
\begin{tabular}{l l} \hline
**Parameter** & **Description** \\ \hline C\({}_{1}\) & The input cost when shipping enterprises choose NESs \\ B\({}_{1}\) & The earnings when shipping enterprises choose TFSs \\ C\({}_{2}\) & The input cost when shipping enterprises choose TFSs \\ B\({}_{2}\) & The earnings when shipping enterprises choose TFSs \\ U\({}_{0}\) & The utility loss of shipping enterprises applying NESs \\ g(m)s & The rewards that shipping enterprises choosing NESs can receive when \\ g(m)s & the government adopts incentive policies to promote NESs \\ g(m) & The coefficient of government incentives \\ s & The initial amount of government rewards \\ T & The tax required to use NESs or TFSs when the government does not \\  & provide tax incentives to use NESs \\ T\({}_{1}\) & The tax required to pay by those who using NESs when the government \\  & provides tax incentives to use NESs \\ T\({}_{2}\) & The tax required to pay is T + T\({}_{2}\) when the government chooses TFSs for \\  & corporate tax T\({}_{2}\) \\ R\({}_{1}\) & The income available to the government when the government does not \\  & adopt the strategy and the enterprise chooses to enter NESs \\  & The additional benefit of the government is (government image \\ R\({}_{2}\) & improvement, etc.) when the government adopts the incentive strategy \\  & and the enterprise chooses NES application \\ R\({}_{3}\) & The original revenue available to the government when enterprises \\  & choose TFSs \\ Variable & \\ x & The probability of the enterprises choosing the new energy ship \\ y & The probability of the government choosing the subsidy \\ \hline \end{tabular}
\end{table} 
Table 1: Description of parameter and variable symbols. \begin{table}
\begin{tabular}{l c c} \hline \hline  & & Government \\ \cline{3-4}  & & Incentive (y) & Non-Incentive (\(1-y\)) \\ \hline \multirow{4}{*}{Shipping enterprises} & NESs & \(\mathrm{B_{1}+g(m)s-C_{1}-}\) \\  & (x) & \(\mathrm{(T-T_{1})-U_{0}}\), \\ \cline{1-1}  & (x) & \(\mathrm{R_{1}+R_{2}+(T-T_{1})-g(m)s}\) \\ \cline{1-1} \cline{2-4}  & TFSs & \(\mathrm{B_{2}-C_{2}-(T+T_{2})}\), \\ \cline{1-1}  & (\(1-x\)) & \(\mathrm{R_{3}+(T+T_{2})}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Payoff matrix. The average expected payoff for shipping enterprises selecting the NESs is as follows:  
\[\rm{E_{s}=xE_{s1}+(1-x)E_{s2}} \tag{3}\]  
The replicator dynamic equation of the shipping enterprise, denoted by x, is shown below:  
\[\rm{F(x)=\frac{dx}{dt}=x(E_{s1}-E_{s})=x(1-x)\{y[g(m)s+T_{1}+T_{2}]+(B_{1}-C_{1} -T-U_{0})-(B_{2}-C_{2}-T)\}} \tag{4}\]  
\[\rm{F^{\prime}(x)=(1-2x)\{y[g(m)s+T_{1}+T_{2}]+(B_{1}-C_{1}-T-U_{0})-(B_{2}-C_{ 2}-T)\}} \tag{5}\]  
From Equation (4), x = 0, x = 1, and y\({}^{*}=\frac{(B_{2}-C_{2}-T)-(B_{1}-C_{1}-T-U_{0})}{g(m)s+T_{1}+T_{2}}\) are roots of \(\rm{F(x)=\frac{dx}{dt}=0}\). According to the dynamic evolutionary stability theory, when \(\rm{F(x)=0}\) and \(\rm{F^{\prime}(x)<0}\), x is the evolutionary stable strategy (ESS). For the shipping enterprise's evolutionary stability strategy, the discussion is given below:  
**Case 1. **_When \(y=y^{*}=\frac{(B_{2}-C_{2}-T)-(B_{1}-C_{1}-T-U_{0})}{g(m)s+T_{1}+T_{2}}\), any \(x\) satisfies \(F(x)\equiv 0\), then any strategy of the shipping enterprise is a stable strategy, and \(y=\frac{(B_{2}-C_{2}-T)-(B_{1}-C_{1}-T-U_{0})}{g(m)s+T_{1}+T_{2}}\) is the probability of the government choosing the incentive policies.