# Exploring and Modeling the Incentive Strategies of New Energy Ship Application in Yangtze River
## 3 Model Analysis
### Analysis of the Evolutionary Stability Strategies of the Government  
The expected payoff of the government incentive policies is as follows:  
\[\mathrm{E_{g1}=x[R_{1}+R_{2}+(T-T_{1})-g(m)s]+(1-x)(R_{3}+T+T_{2})} \tag{6}\]  
The expected payoff of the government without incentive policies is as follows:  
\[\mathrm{E_{g2}=x(R_{1}+T)+(1-x)(R_{3}+T)} \tag{7}\]  
The average expected government return is as follows:  
\[\mathrm{E_{g}=yE_{g1}+(1-y)E_{g2}} \tag{8}\]  
The replicators' dynamic equation of government, represented by F(y), is as below:  
\[\mathrm{F(y)=\frac{dy}{dt}=y\big{(}E_{g1}-E_{g}\big{)}=y(1-y)\{x[R_{2}-T_{1}-T _{2}-g(m)s]+T_{2}\} \tag{9}\]  
\[\mathrm{F(y)=(1-2y)\{x[R_{2}-T_{1}-T_{2}-g(m)s]+T_{2}\} \tag{10}\]  
It is clear that \(y=0,\ y=1\), and \(x^{*}=\frac{T_{2}}{g(m)s+T_{1}+T_{2}-R_{2}}\) are three roots of \(\mathrm{F(y)=0}\). Moreover, when \(\mathrm{F(y)=0}\) and \(\mathrm{F^{\prime}(y)<0}\), \(y\) is the government's stability strategy. The discussion is given below:  
**Case 5**.: _When \(x=x^{*}=\frac{T_{2}}{g(m)s+T_{1}+T_{2}-R_{2}}\), and any \(y\) satisfies \(F(y)\equiv 0\), then the government's arbitrary choice strategy is a stable strategy._  
When \(x\
eq x^{*}\
eq\frac{T_{2}}{g(m)s+T_{1}+T_{2}-R_{2}}\), there are also three cases:  
**Case 6**.: _When \(x^{*}=\frac{T_{2}}{g(m)s+T_{1}+T_{2}-R_{2}}<0\), \(g(m)s+T_{1}+T_{2}<R_{2}\), \(x>\frac{T_{2}}{g(m)s+T_{1}+T_{2}-R_{2}}\), and \(x[R_{2}-T_{1}-T_{2}-g(m)s]+T_{2}>0\). When \(y=0\), \(y=1\), \(F(0)=0\), \(F^{\prime}(0)>0\), \(F(1)=0\), and \(F^{\prime}(1)<0\), \(y=1\) is the only ESS. Case 6 shows that when the additional income obtained by the government by adopting the incentive policies is greater than the reward amount paid, the government chooses the incentive strategy when stabilizing the incentive strategy._  
**Case 7**.: _When \(0<x^{*}=\frac{T_{2}}{g(m)s+T_{1}+T_{2}-R_{2}}<1\), there is \(R_{2}<g(m)s+T_{1}\). Another two situations are discussed as follows: If \(x<x^{*}=\frac{T_{2}}{g(m)s+T_{1}+T_{2}-R_{2}}\), \(F(0)=0\), \(F^{\prime}(0)>0\), \(F(1)=0\), and \(F^{\prime}(1)<0\), then \(y=1\) is the only ESS; if \(x>x^{*}=\frac{T_{2}}{g(m)s+T_{1}+T_{2}-R_{2}}\), \(F(0)=0\), \(F^{\prime}(0)<0\), \(F(1)=0\), and \(F^{\prime}(1)>0\), then \(y=0\) is the only ESS. Case 7 shows that when the probability of an enterprise choosing NESs is lower than \(\frac{T_{2}}{g(m)s+T_{1}+T_{2}-R_{2}}\), the behavior of  
Figure 2: Phase diagrams for shipping enterprises’ strategies.  
the government will change from non-incentives to incentives; and when the probability of choosing NESs is higher than \(\frac{T_{2}}{g(m)s+T_{1}+T_{2}-R_{2}}\), the government will change from incentives to non-incentives._  
The dynamic evolutionary path of Case 7 is given in Figure 3.  
**Case 8**.: _When \(x^{*}=\frac{T_{2}}{g(m)s+T_{1}+T_{2}-R_{2}}>1\), there is \(g(m)s+T_{1}<R_{2}<g(m)s+T_{1}+T_{2}\), there is any \(x<\frac{T_{2}}{g(m)s+T_{1}+T_{2}-R_{2}}\), and \(x[R_{2}-T_{1}-T_{2}-g(m)s]+T_{2}>0\), then \(F(0)=0\), \(F(0)>0\), \(F(1)=0\), and \(F^{\prime}(1)<0\), so \(y=1\) is the only ESS. Case 8 shows that when the government adopts the subsidy policy and the additional earnings is between \(g(m)s+T_{1}\) and \(g(m)s+T_{1}+T_{2}\), the government chooses the incentive strategy._