## 4 Model Stability Analysis
### Analysis of Creators
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Platforms**} & \multicolumn{3}{c}{**Regulators**} \\ \cline{3-4} \multicolumn{2}{c}{} & \multicolumn{1}{c}{\(\gamma_{1}\) (\(z\))} & \multicolumn{1}{c}{\(\gamma_{2}\) (\(1-z\))} \\ \hline \multirow{5}{*}{Creators} & \(\alpha_{1}\) (\(x\)) & \(\beta_{1}\) (\(y\)) & \((R_{\mathrm{c}}-C_{\mathrm{co}},R_{\mathrm{P}}+M_{\mathrm{p}},-C_{\mathrm{g}}-M_{ \mathrm{p}}+\) & \((R_{\mathrm{c}}-C_{\mathrm{co}},R_{\mathrm{P}},R_{\mathrm{g}})\) \\ \cline{3-4}  & \(\beta_{2}\) (\(1-y\)) & \((R_{\mathrm{c}}-C_{\mathrm{co}},R_{\mathrm{P}}-C_{\mathrm{p}}-F_{\mathrm{p}},-C_{ \mathrm{g}}+F_{\mathrm{P}}\) & \((R_{\mathrm{c}}-C_{\mathrm{co}},R_{\mathrm{P}}-C_{\mathrm{p}},R_{\mathrm{g}})\) \\ \cline{2-4}  & \(\alpha_{2}\) (\(1-x\)) & \(\beta_{1}\) (\(y\)) & \((-C_{\mathrm{cf}}-C_{\mathrm{cfs}}-F_{\mathrm{c}},R_{\mathrm{P}}+M_{\mathrm{p}},-C_{ \mathrm{g}}+F_{\mathrm{c}}-M_{\mathrm{p}})\) & \((-C_{\mathrm{cf}}-C_{\mathrm{cfs}},R_{\mathrm{P}},0)\) \\ \cline{2-4}  & \(\beta_{2}\) (\(1-y\)) & \((R_{\mathrm{c}}-C_{\mathrm{cf}}-C_{\mathrm{cfs}}-C_{\mathrm{cfr}}-F_{\mathrm{c }},R_{\mathrm{P}}-\) & \((R_{\mathrm{c}}-C_{\mathrm{cf}}-C_{\mathrm{cfs}}-C_{\mathrm{cfr}},R_{\mathrm{P}}-\) \\ \cline{2-4}  & \(C_{\mathrm{p}}+C_{\mathrm{cfr}}-F_{\mathrm{p}},-C_{\mathrm{g}}+F_{\mathrm{P}}+F_{ \mathrm{c}}-D_{\mathrm{g}})\) & \(C_{\mathrm{p}}+C_{\mathrm{cfr}},-D_{\mathrm{g}}-T_{\mathrm{g}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Payoff matrix among the regulators, platforms, and creators. \[F^{\prime}(x)=(1-2x)((R_{\rm c}-C_{\rm cfr})y+zF_{\rm c}+C_{\rm cf}+C_{\rm cfr}+C_{ \rm cfs}-C_{\rm co}) \tag{4}\]  
\[{\rm Set}\ G(y)=(R_{\rm c}-C_{\rm cfr})y+zF_{\rm c}+C_{\rm cf}+C_{\rm cfr}+C_{ \rm cfs}-C_{\rm co} \tag{5}\]  
According to the stability theorem for replicating dynamic equations, the corresponding strategy is ESS only when \(x\) satisfies \(F(x)=0\) and \(F^{\prime}(x)<0\)[52]. 1. When \(y=(-zF_{\rm c}-C_{\rm cf}-C_{\rm cfr}-C_{\rm cfs}+C_{\rm co})/(R_{\rm c}-C_{ \rm cfr})=y^{\star}\), there are \(G(y)=0\), and \(F(x)\equiv 0\); the digital content creator's choice of any proportion of original strategy is stable strategy, meaning that their strategic choice will not change with the passage of time;
2. Due to \(G(y)\) being a monotonically increasing function, when \(y<y^{\star}\), there are \(G(y)<0\), \(F^{\prime}(1)>0\), and \(F^{\prime}(0)<0\); therefore, \(x=0\) is the ESS of the digital content creator. When \(y>y^{\star}\), there are \(G(y)>0\), \(F^{\prime}(1)<0\), and \(F^{\prime}(0)>0\); therefore, \(x=1\) is ESS of the digital content creator. The evolutionary trend of digital content creators' strategy is shown in Figure 2. The probability that the creator steadily chooses the non-original strategy is \(V_{\rm A1}\), that is, the volume of \({\rm A}_{1}\), and the probability that steadily chooses the original strategy is \(V_{\rm A2}\), that is, the volume of \({\rm A}_{2}\). The expression is as follows:  
\[V_{\rm A1}=\int_{0}^{1}\int_{0}^{1}\frac{-zF_{\rm c}-C_{\rm cf}-C_{\rm cfr}- C_{\rm cfs}+C_{\rm co}}{R_{\rm c}-C_{\rm cfr}}dxdz=\frac{2(C_{\rm co}-C_{\rm cf}-C_{ \rm cfr}-C_{\rm cfs})-F_{\rm c}}{2(R_{\rm c}-C_{\rm cfr})} \tag{6}\]  
\[V_{\rm A2}{=1-V_{\rm A1}=\frac{2(R_{\rm c}-C_{\rm co}+C_{\rm cf}+C_{\rm cfs}) +F_{\rm c}}{2(R_{\rm c}-C_{\rm cfr})}} \tag{7}\]  
**Lemma 1**. : _The probability of digital content creators choosing the original strategy is positively correlated with the fines, sales revenue, collusive costs, and speculative costs they bear while negatively correlating with the production costs saved by choosing a fake strategy._  
Proof.