## 3 The Stackelberg Game Models  
### P-Model: A Stackelberg Game Model with Platform as Leader  
The stackelberg game model with platform as leader refers to that, among the participants of the game between the advertising platform and the advertiser, the advertising platform is more powerful in the market power structure and controlling the cooperation between the two parties. Therefore, the platform as a leader dominates the whole game, while the advertiser is in a subordinate position. It can be considered as a typical stackelberg master-slave game [16]. In addition, we call it as P-model in short. As a leader, the platform determines the proportion of the advertiser's transaction commission, and the cost of brand advertising effort paid by the platform. In addition, the advertiser determines the optimal price of advertising products and the cost of advertising effort paid by the advertiser. To obtain the solution to the structure, the platform's decision problem is solved based on the advertiser's response [16; 18]. Therefore, the profit of the e-commerce platform is as the follow (9):  
\[ \begin{split}\prod_{p}&=\lambda p_{a}Q\big{(}p_{a}, B_{a},B_{p}\big{)}-B_{p}\\ &=\lambda p_{a}(q-\mu p_{a})\left(1+\sqrt{\frac{2B_{a}}{\gamma_{ a}}}+\sqrt{\frac{2B_{p}}{\gamma_{p}}}\right)-B_{p}\end{split}  \tag{9}\]  
The profit of the advertiser is calculated as below (10):  
\[ \begin{split}\prod_{a}&=(p_{a}-c_{a})Q\big{(}p_{a}, B_{a},B_{p}\big{)}-\lambda p_{a}Q\big{(}p_{a},B_{a},B_{p}\big{)}-B_{a}\\ &=[(1-\lambda)p_{a}-c_{a}](q-\mu p_{a})\left(1+\sqrt{\frac{2B_{a }}{\gamma_{a}}}+\sqrt{\frac{2B_{p}}{\gamma_{p}}}\right)-B_{a}\end{split}  \tag{10}\]  
The profit of the system is the following (11):  
\[ \begin{split}\prod&=\prod_{p}+\prod_{a}\\ &=(p_{a}-c_{a})(q-\mu p_{a})\left(1+\sqrt{\frac{2B_{a}}{\gamma_{ a}}}+\sqrt{\frac{2B_{p}}{\gamma_{p}}}\right)-B_{p}-B_{a}\end{split}  \tag{11}\]  
This stackelberg game model is a typical dynamic game of complete information. For the platform, the decision variables are the income commission proportion \(\lambda\) for platform due to advertisers'transaction and the effort cost of the platform's brand advertising \(B_{p}\). For the advertiser, the decision variables are the price of the advertising product \(p_{a}\) and the effort cost of the advertiser's brand advertising \(B_{a}\). It can be solved by the backward induction method to find the stackelberg equilibrium. **Proposition 1**. : _In the stackelberg P-model, the optimal decision variable \(\lambda^{p*}\) of the e-commerce platform is (12):_  
\[\lambda^{p*}=1+\frac{\mu^{2}{c_{a}}^{2}}{q\sqrt[3]{3\mu^{2}{c_{a}}^{2}\left[9q -\sqrt{3\mu^{2}{c_{a}}^{2}+81q^{2}}\right]}}-\frac{1}{3}\sqrt[3]{3\mu^{2}{c_{a }}^{2}\left[9q-\sqrt{3\mu^{2}{c_{a}}^{2}+81q^{2}}\right]} \tag{12}\]  
_Then, the other optimal decision variables \(B_{p}^{p*}\) is as the follow (13):_  
\[B_{p}^{p*}=\frac{\lambda^{p*2}\left[\left(1-\lambda^{p*}\right)^{2}q^{2}-\mu^ {2}{c_{a}}^{2}\right]^{2}}{32\mu^{2}\gamma_{p}\left(1-\lambda^{p*}\right)^{4}} \tag{13}\]  
**Proposition 2**. : _In the stackelberg P-model, the optimal decision variables of the advertiser are (14) and (15):_  
\[{p_{a}}^{p*}=\frac{\left(1-\lambda^{p*}\right)q+\mu{c_{a}}}{2\mu(1-\lambda^{ p*})} \tag{14}\]  
\[{B_{a}}^{p*}=\frac{\left[\left(1-\lambda^{p*}\right)q-\mu{c_{a}}\right]^{4}}{3 2\gamma_{a}\mu^{2}(1-\lambda^{p*})^{2}} \tag{15}\]  
**Proposition 3**. : _In the stackelberg P-model, the e-commerce platform's and advertiser's maximum profits are then, respectively, in the Equations (16) and (17):_  
\[\begin{array}{ll}{\prod_{p}}^{p*}&=\left(\frac{\frac{4}{7\mu} \lambda^{p*3}q^{2}-\frac{6}{7\gamma}\lambda^{p*3}q^{2}+4\frac{8}{7\gamma}\mu \lambda^{p*2}q{c_{a}}+16\mu\lambda^{p*2}}{64\mu^{2}(1-\lambda^{p*})^{4}}\right.