## 4 Model Stability Analysis
### Analysis of Platforms  
The expected return for platforms that choose strategy \(\beta_{1}\) is \(U_{\rm p1}\). The expected return for platforms that choose strategy \(\beta_{2}\) is \(U_{\rm p2}\). According to the income matrix, it can be concluded that  
\[ \begin{array}{c}U_{\rm p1}=xz(R_{\rm P}+M_{\rm P})+x(1-z)(R_{\rm P})+(1-x)z( R_{\rm P}+M_{\rm P})+(1-x)(1-z)(R_{\rm P})\\ =zM_{\rm P}+R_{\rm P}\end{array}  \tag{8}\]  
\[ \begin{array}{c}U_{\rm p2}=xz(R_{\rm P}-C_{\rm P}-F_{\rm P})+x(1-z)(R_{\rm P }-C_{\rm P})+(1-x)z(R_{\rm P}-C_{\rm P}+C_{\rm cfr}-F_{\rm P})+(1-x)(1-z)(R_{ \rm P}-C_{\rm P}+C_{\rm cfr})\\ =-xC_{\rm cfr}-zF_{\rm P}+C_{\rm cfr}-C_{\rm P}+R_{\rm P}\end{array}  \tag{9}\]  
The replication dynamic equation of the NFT publishing platform and its first derivative with respect to \(y\) are as follows:  
\[ \begin{array}{c}F(y)=y(1-y)(U_{\rm P1}-U_{\rm p2})=y(1-y)(zM_{\rm P}+R_{\rm P }-(-xC_{\rm cfr}-zF_{\rm P}+C_{\rm cfr}-C_{\rm P}+R_{\rm P}))\\ =y(1-y)(xC_{\rm cfr}+z(F_{\rm P}+M_{\rm P})-C_{\rm cfr}+C_{\rm P})\end{array}  \tag{10}\]  
\[ \begin{array}{c}F^{\prime}(y)=(1-2y)(xC_{\rm cfr}+z(F_{\rm P}+M_{\rm P})-C_ {\rm cfr}+C_{\rm P})\end{array}  \tag{11}\]  
\[ \begin{array}{c}\mbox{Set. }H(z)=xC_{\rm cfr}+z(F_{\rm P}+M_{\rm P})-C_{\rm cfr}+C_{ \rm P}\end{array}  \tag{12}\]  
1. When \(z=((1-x)C_{\rm cfr}-C_{\rm P})/(F_{\rm P}+M_{\rm P})=z^{\star}\), there are \(H(z)=0\), and \(F(y)\equiv 0\); the NFT publishing platform's choice of any proportion of fair detection strategy is stable strategy;
2. Due to \(H(z)\) being a monotonically increasing function, when \(z<z^{\star}\), there are \(H(z)<0\), \(F^{\prime}(1)>0\), and \(F^{\prime}(0)<0\); therefore, \(y=0\) is the ESS of the NFT publishing platform. When \(z>z^{\star}\), there are \(H(z)>0\), \(F^{\prime}(1)<0\), and \(F^{\prime}(0)>0\); therefore, \(y=1\) is the ESS of the NFT publishing platform. The evolutionary trend of the NFT service platforms' strategy is shown in Figure 3. The tangent crosses the point \(((C_{\rm cfr}-C_{\rm P})/C_{\rm cfr}\), \(0\), \(0\)), and \(0<(C_{\rm cfr}-C_{\rm P})/C_{\rm cfr}<1\). The probability that the NFT publishing platform steadily chooses the fair detection strategy is \(V_{\rm B1}\), which is the volume of \({\rm B}_{1}\), and the probability that steadily chooses the collusion detection strategy is \(V_{\rm B2}\), which is the Volume of \({\rm B}_{2}\). The expression is as follows:  
\[ \begin{array}{c}V_{\rm B2}=\int_{0}^{1}\int_{0}^{(C_{\rm cfr}-C_{\rm P})/C_ {\rm cfr}}\frac{(1-x)C_{\rm cfr}-C_{\rm P}}{F_{\rm P}+M_{\rm P}}dxdy=\frac{ \left(C_{\rm cfr}-C_{\rm P}\right)^{2}}{2C_{\rm cfr}\left(F_{\rm P}+M_{\rm P} \right)}\end{array}  \tag{13}\]  
**Lemma 3**: _The probability of the NFT digital publishing platform choosing fair detection is negatively correlated with the collusive costs and positively correlated with the fines faced, rewards received, and speculative costs paid._  
**Proof 3**: _According to the expression for \(V_{\text{B1}}\) (probability that the platform chooses the fair detection strategy), solving for the first-order partial derivatives of each parameter, obtain \(\partial V_{\text{B1}}/\partial C_{\text{cfr}}<0\), \(\partial V_{\text{B1}}/\partial M_{\text{p}}>0\), \(\partial V_{\text{B1}}/\partial F_{\text{p}}>0\), \(\partial V_{\text{B1}}/\partial C_{\text{p}}>0\). \(\square\)_  
Lemma 3 indicates that when the collusion revenue is significant, regulators should choose a strict supervision strategy and increase the intensity of rewards and fines. At the same time, regulators can also increase the speculative costs of complicity by enhancing the professionalism of testers and expanding media disclosure to promote fair testing.