## 4 Model Stability Analysis
### Analysis of Regulators  
The expected return for regulators who choose strategy \(\gamma_{1}\) is \(U_{\text{g1}}\). The expected return for regulators who choose strategy \(\gamma_{2}\) is \(U_{\text{g2}}\). According to the income matrix, it can be concluded that  
\[U_{\text{g1}}=xy(-C_{\text{g}}-M_{\text{p}}+R_{\text{g}})+x(1-y)(-C_{\text{g}  
\[ \begin{split} U_{\rm{g2}}&=xy(R_{\rm{g}})+x(1-y)(R_{\rm{g} })+(1-x)y(0)+(1-x)(1-y)(-D_{\rm{g}}-T_{\rm{g}})\\ &=x(D_{\rm{g}}+R_{\rm{g}}+T_{\rm{g}})+y(D_{\rm{g}}+T_{\rm{g}})+ xy(-D_{\rm{g}}-T_{\rm{g}})-D_{\rm{g}}-T_{\rm{g}}\end{split}  \tag{16}\]  
The replication dynamic equation of regulators and its first derivative with respect to \(z\) are as follows:  
\[ \begin{split} F(z)=z(1&-z)(U_{\rm{g1}}-U_{\rm{g2}}) =z(1-z)(x(D_{\rm{g}}-F_{\rm{c}}+R_{\rm{g}})+y(D_{\rm{g}}-F_{\rm{p}}-M_{\rm{p}}) -xyD_{\rm{g}}-C_{\rm{g}}-D_{\rm{g}}\\ &+F_{\rm{c}}+F_{\rm{p}}-(x(D_{\rm{g}}+R_{\rm{g}}+T_{\rm{g}})+y(D _{\rm{g}}+T_{\rm{g}})+xy(-D_{\rm{g}}-T_{\rm{g}})-D_{\rm{g}}-T_{\rm{g}}))\\ &=z(1-z)(x(-F_{\rm{c}}-T_{\rm{g}})+y(-F_{\rm{p}}-M_{\rm{p}}-T_{ \rm{g}})+xyT_{\rm{g}}-C_{\rm{g}}+F_{\rm{c}}+F_{\rm{p}}+T_{\rm{g}})\\ \end{split}  \tag{17}\]  
\[ \begin{split} F^{\prime}(z)=(1-2z)(x(-F_{\rm{c}}-T_{\rm{g}})+y(- F_{\rm{p}}-M_{\rm{p}}-T_{\rm{g}})+xyT_{\rm{g}}-C_{\rm{g}}+F_{\rm{c}}+F_{\rm{p}}+T_{ \rm{g}})\end{split}  \tag{18}\]  
\[ \begin{split}\text{Set. }J(y)=x(-F_{\rm{c}}-T_{\rm{g}})+y(-F_{\rm{p}}-M_{ \rm{p}}-T_{\rm{g}})+xyT_{\rm{g}}-C_{\rm{g}}+F_{\rm{c}}+F_{\rm{p}}+T_{\rm{g}} \end{split}  \tag{19}\]  
When \(y=(-C_{\rm{g}}+F_{\rm{c}}+F_{\rm{p}}+T_{\rm{g}}-x(F_{\rm{c}}+T_{\rm{g}}))/(F_ {\rm{p}}+M_{\rm{p}}+T_{\rm{g}}-xT_{\rm{g}})=y^{\star\star}\), there are \(J(y)=0\), and \(F(z)\equiv 0\), the regulator's choice of any proportion of strict strategy is stable strategy;  
Due to \(J(y)\) being a monotonically decreasing function, when \(y<y^{\star\star}\), there are \(J(y)>0\), \(F^{\prime}(1)<0\), and \(F^{\prime}(0)>0\); therefore, \(z=1\) is the ESS. When \(y>y^{\star\star}\), there are \(J(y)<0\), \(F^{\prime}(1)>0\), and \(F^{\prime}(0)<0\); therefore, \(z=0\) is the ESS of the NFT publishing platform. The evolutionary trend of the regulators' strategy is shown in Figure 4. The probability that regulators steadily choose a strict supervision strategy is \(V_{\rm{C1}}\), that is, the volume of \(\rm{C_{1}}\), and the probability that steadily chooses the loss supervision strategy is \(V_{\rm{C2}}\), that is, the volume of \(\rm{C_{2}}\). The expression is as follows:  
\[ \begin{split} V_{\rm{C1}}=\int_{0}^{1}\int_{0}^{1}\frac{-C_{\rm{g }}+F_{\rm{c}}+F_{\rm{p}}T_{\rm{g}}-x\big{(}F_{\rm{c}}+T_{\rm{g}}\big{)}}{F_{ \rm{p}}+M_{\rm{p}}+T_{\rm{g}}-xT_{\rm{g}}}dxdz=\frac{F_{\rm{c}}+T_{\rm{g}}}{T_ {\rm{g}}}-\Bigg{(}\frac{C_{\rm{g}}+M_{\rm{p}}}{T_{\rm{g}}}+\frac{F_{\rm{c}} \big{(}F_{\rm{p}}+M_{\rm{p}}\big{)}}{T_{\rm{g}}{}^{2}}\Bigg{)}\text{ln}(1+ \frac{T_{\rm{g}}}{F_{\rm{p}}+M_{\rm{p}}})\end{split}  \tag{20}\]  
\[ \begin{split} V_{\rm{C2}}=1-V_{\rm{C1}}=\Bigg{(}\frac{C_{\rm{g}}+M _{\rm{p}}}{T_{\rm{g}}}+\frac{F_{\rm{c}}\big{(}F_{\rm{p}}+M_{\rm{p}}\big{)}}{T_ {\rm{g}}{}^{2}}\Bigg{)}\text{ln}(1+\frac{T_{\rm{g}}}{F_{\rm{p}}+M_{\rm{p}}})- \frac{F_{\rm{c}}}{T_{\rm{g}}}\end{split}  \tag{21}\]  
**Lemma 5**. : _The probability of strict supervision by regulators is positively related to the fines borne by creators and the penalties suffered by regulatory authorities for loose supervision, negatively related to the rewards received by NFT service platforms, and the relationship with the fines borne by NFT service platforms is influenced by multiple factors._  
Proof. : According to the expression for \(V_{\rm{C1}}\) (probability that regulators chooses strict supervision), obtain \(\partial V_{\rm{C1}}/\partial F_{\rm{c}}>0\), \(\partial V_{\rm{C1}}/\partial T_{\rm{g}}>0\), \(\partial V_{\rm{C1}}/\partial M_{\rm{p}}<0\), \(\partial V_{\rm{C1}}/\partial F_{\rm{p}}>0(\text{s.t. }C_{\rm{g}}+M_{\rm{p}}>F_{ \rm{c}})\). \(\square\)  
Figure 4: The evolutionary trend of regulators’ strategy.