## 3 The Stackelberg Game Models
### P-Model: A Stackelberg Game Model with Platform as Leader
\\ &\left.+\frac{\frac{12}{7\gamma}\lambda^{p*2}q^{2}-\frac{12}{7\mu}\lambda^{p*2 }q^{2}-\frac{16}{7\mu}\lambda^{p*q}{c_{a}}+\frac{6}{7\gamma}\mu^{2}{c_{a}}^{2 }}{64\mu^{2}(1-\lambda^{p*})^{4}}\right.\\ &\left.+\frac{16\mu+\frac{4}{7\mu}\lambda^{p*}{c_{a}}^{2}+\frac{8}{7\mu}\mu{c_ {a}}-\frac{6}{7\gamma}\mu^{p*2}q^{2}-32\mu\lambda^{p*}}{64\mu^{2}(1-\lambda^{ p*})^{4}}\right.\\ &\left.+\frac{\frac{12}{7\mu}\lambda^{p*}q^{2}-\frac{4}{7\mu}{c_{a}}^{2}-\frac{ 4}{7\mu}{c_{a}}^{2}-\frac{4}{7\mu}{c_{a}}^{2}}{64\mu^{2}(1-\lambda^{p*})^{4}} \right)\cdot\frac{\left[\left(1-\lambda^{p*}\right)^{2}q^{2}-\mu^{2}{c_{a}}^{2 }\right]}{64\mu^{2}(1-\lambda^{p*})^{4}}\\ &\left.+\frac{\frac{12}{7\mu}\lambda^{p*2}q^{2}-\frac{4}{7\mu}{c_{a}}^{2}-\frac {4}{7\mu}{c_{a}}^{2}-\frac{4}{7\mu}{c_{a}}^{2}+\frac{6}{7\mu}{c_{a}}^{2}}{64 \mu^{2}(1-\lambda^{p*})^{3}}\right.\\ &\left.+\frac{\frac{12}{7\mu}\lambda^{p*3}q^{2}-\frac{6}{7\mu}{c_{a}}^{2}{2}{c _{a}}^{2}}{64\mu^{2}(1-\lambda^{p*})^{3}}\right)\cdot\frac{\left[\left(1- \lambda^{p*}\right)^{2}q^{2}-\mu^{2}{c_{a}}^{2}\right]}{64\mu^{2}(1-\lambda^{p *})^{4}}\\ &\left.+\frac{12}{7\mu}\lambda^{p*3}q^{2}-\frac{6}{7\mu}{c_{a}}^{2}{2}{c_{a}} ^{2}+\frac{6}{7\mu}{c_{a}}^{2}{2}{c_{a}}^{2}-\frac{18}{7\mu}{c_{a}}^{2}{2}{c _{a}}^{2}}{64\mu^{2}(1-\lambda^{p*})^{3}}\\ &\left.+\frac{_And the total profit in the advertising cooperation is calculated by Equation (18):_  
\[ \begin{array}{ll}\prod^{p*}&=\left[\frac{(1-\lambda^{p*})q+\mu c_{a}}{2\mu(1- \lambda^{p*})}-c_{a}\right]\cdot\left[q-\frac{(1-\lambda^{p*})q+\mu c_{a}}{2(1- \lambda^{p*})}\right]\\ &\cdot\left[1+\frac{[(1-\lambda^{p*})q-\mu c_{a}]^{2}}{4\gamma_{2}\mu^{2}(1- \lambda^{p*})}+\frac{\lambda^{p*}\left[(1-\lambda^{p*})^{2}q^{2}-\mu^{2}c_{a} ^{2}\right]}{4\gamma_{p}\mu(1-\lambda^{p*})^{2}}\right]\\ &-\frac{\lambda^{p*2}\left[(1-\lambda^{p*})^{2}q^{2}-\mu^{2}c_{a} \right]^{2}}{32\gamma_{p}\mu^{2}(1-\lambda)^{4}}\\ &-\frac{\left[(1-\lambda^{p*})q^{2}-\mu c_{a}\right]^{4}}{32\gamma_{4}\mu^{2}( 1-\lambda)^{2}}\end{array}  \tag{18}\]  
**Proof. ** Firstly, for any given platform's decision variables \(B_{p}\) and \(\lambda\), the optimal decision variables \(B_{a}\) and \(p_{a}\) of the advertiser are to be determined. Therefore, the profit function of the advertiser on the premise of maximizing his own interests is derived from which the solution to the optimization problem can be obtained as shown in Equation (19). \[\max_{p_{a},B_{a}}\prod_{a}=[(1-\lambda)p_{a}-c_{a}](q-\mu p_{a})\left(1+\sqrt{ \frac{2B_{a}}{\gamma_{a}}}+\sqrt{\frac{2B_{p}}{\gamma_{p}}}\right)-B_{a} \tag{19}\]  
_s.t._\(0<p_{a}<\frac{q}{p}\), \(B_{a}>0\)  
Because \(\prod_{a}\) is a strict concave function of the price and the cost of advertising effort. Let \(\frac{q\prod_{a}}{q\beta_{a}}=0\)  
We derive (20):  
\[{B_{a}}^{p*}=\frac{1}{2\gamma_{a}}[(1-\lambda)p_{a}-c_{a}]^{2}(q-\mu p_{a})^{2} \tag{20}\]  
Let \(\frac{q\prod_{a}}{q\beta_{a}}=0\)  
We obtain (14):  
\[{p_{a}}^{p*}=\frac{(1-\lambda)q+\mu c_{a}}{2\mu(1-\lambda)}\]  
If \(B_{a}^{p*}\) is represented in the form of \(p_{a}p^{*}\), then we get (15):  
\[{B_{a}}^{p*}=\frac{[(1-\lambda)q-\mu c_{a}]^{4}}{32\gamma_{a}\mu^{2}(1-\lambda )^{2}}\]  
Based on the idea of the backward induction, the platform determines his decision variables according to the optimal decisions of the advertiser, namely, the optimal cost of brand advertising effort and the optimal price of advertising product. Then, to maximize the platform's own interests, the profit function and the solution are derived as follows (21). \[\max_{\lambda,B_{p}}\prod_{p}=\lambda p_{a}(q-\mu p_{a})\left(1+\sqrt{\frac{2B _{a}}{\gamma_{a}}}+\sqrt{\frac{2B_{p}}{\gamma_{p}}}\right)-B_{p} \tag{21}\]  
_s.t._\(0<\lambda<1,B_{p}>0\)  
In the form of \({p_{a}}^{p*}\) and \({B_{a}}^{p*}\), \(\max_{\lambda,B_{p}}\prod_{p}\) can be represented in Equation (22):  
\[