## 3 The Stackelberg Game Models
### A-model: A Stackelberg Game Model with Advertiser as Leader  
Compared with the advertising platform, the advertiser is much stronger in the market power structure and more effectively controls the cooperation of both sides in this kind of cases. As a result, the e-commerce platform is willing to be in a subordinate position to attract such big advertisers. This is a typical stackelberg master-slave game with advertiser as leader. In addition, we call it as A-model. Since the A-model of advertiser leading brand effect is a complete information dynamic game, for the advertiser, the decision variable is the quality cost of the advertiser's brand advertising and the pricing of the advertiser's advertising products. For the platform, the decision variable is the quality cost of the platform's brand advertising and the proportion of the platform's transaction commission to the advertiser's advertising products. The advertiser, as the leader, first determines the pricing of his own advertising products and the corresponding cost of brand advertising quality. Then the platform determines the proportion of the transaction commission charged by the advertiser and his cost of brand advertising quality. **Proposition 4**. : _In the stackelberg A-model, the optimal decision variable \({p_{a}}^{a*}\) of the advertiser is (23):_  
\[{p_{a}}^{a*}=\frac{q}{2\mu}+\frac{c_{a}}{2} \tag{23}\]  
_Then, the other optimal variable \({B_{a}}^{a*}\)is as the follow (24):_  
\[{B_{a}}^{a*}=\frac{8\Big{(}4+\frac{6}{\gamma_{p}}{p_{a}}^{a*2}+\frac{6}{ \gamma_{p}}{c_{a}}q-\frac{6}{\gamma_{p}}{p_{a}}^{a*}{c_{a}}-\frac{6}{\gamma_{p }}{p_{a}}^{a*}q\Big{)}^{2}}{\gamma_{a}\Big{(}\frac{16}{\gamma_{a}}+\frac{18}{ \gamma_{p}}\Big{)}^{2}} \tag{24}\]  
**Proposition 5**. : _In the stackelberg A-model, the optimal decision variables of the e-commerce platform are (25) and (26):_  
\[\lambda^{a*}=\frac{2\Big{(}\frac{16}{\gamma_{a}}+\frac{18}{\gamma_{p}}\Big{)} +\frac{8}{\gamma_{a}}\Big{[}\frac{6}{\gamma_{p}}(q-\mu{p_{a}}^{a*})({p_{a}}^{ a*}-{c_{a}})-4\Big{]}}{\frac{6}{\gamma_{p}}{p_{a}}^{a*}(q-\mu{p_{a}}^{a*})\Big{(} \frac{16}{\gamma_{a}}+\frac{18}{\gamma_{p}}\Big{)}} \tag{25}\]  
\[{B_{p}}^{a*}=\frac{\frac{2}{\gamma_{p}}\Big{[}\frac{4}{\gamma_{a}}({p_{a}}^{a* }-{c_{a}})(q-\mu{p_{a}}^{a*})+3\Big{]}^{2}}{\Big{(}\frac{16}{\gamma_{a}}+\frac {18}{\gamma_{p}}\Big{)}^{2}} \tag{26}\]  
**Experiment 2:**_the corresponding variables are given as below._  
\[q=1000,\ c_{a}=100,\ \mu=8,\ \gamma_{p}=40,\ \gamma_{a}=60\]  
_The optimal equilibriums and profits of the two parties in the different models are calculated and displayed in Table 2._  
**Experiment 3:**_the corresponding variables are given as below._  
\[q=1000,\ c_{a}=200,\ \mu=4,\ \gamma_{p}=60,\ \gamma_{a}=40\]  
_Then the optimal equilibriums and profits of the platform and the advertiser are shown in Table 3._  
By the comparisons of the results in the above experiments, we make some analysis as below:  
1. The income commission proportion for e-commerce platform due to advertisers' transaction would be higher in the P-model than that in the A-model. Because in the P-model, the e-commerce platform plays a leading role in decision-making and is willing to enhance the proportion to maximize his own interest. 2. The price of the advertiser's product is higher in the P-model than that in the A-model. Because in the P-model, the advertiser is in the passive position. To ensure his own profit, it should make a relative high price since the platform makes a higher commission proportion in P-model than in A-model.