## 2 Descriptions and Hypothesis  
For parsimony, this paper deals with simple advertising mode having only one e-commerce platform and one advertiser. The relevant model description and basic hypothesis are given below:  
\(\xi_{0}\) The original growth contribution rate at which the demand of products enhances due to consumer'recognition without brand advertising on the e-commerce platform  
\(\xi_{1}\) The derived growth contribution rate at which the demand of products enhances because of the consumers' recognition with brand advertising on the e-commerce platform  
\(\xi\) The total growth contribution rate at which the demand of products enhances influenced by the consumers' recognition of commodities through brand advertising channels and non-brand advertising channels on the e-commerce platform  
\(q\) The aggregate demand for advertiser's products without brand advertising on the e-commerce platform  
\(p_{a}\) The advertising product price  
\(\lambda\) The income commission proportion for e-commerce platform due to advertisers' transaction  
\(B_{p}\) The cost of advertising effort of the platform brand  
\(B_{a}\) The cost of advertising effort of the advertiser brand  
\(\mu\) The sensitivity coefficient of demand to price  
\(E_{p}\) The advertising efficacy of the platform brand  
\(E_{a}\) The advertising efficacy of the advertiser brand  
\(\gamma_{p}\) The coefficient of cost to effect of the platform brand  
\(\gamma_{a}\) The coefficient of cost to effect of the advertiser brand  
\(c_{a}\) The cost of the advertiser's productsIn the game model, the decision variables of the e-commerce platform are \(\lambda\) and \(B_{p}\), and the decision variables of the advertiser are \(p_{a}\) and \(B_{a}\). Since \(\lambda\) is the income commission proportion for platform due to advertisers' transaction, it largely determines the profit of the platform. In addition, the cost of advertising effort of the platform brand \(B_{p}\) decides the advertising efficacy of the platform brand. \(p_{a}\) is the advertising product price which directly affects the income of the advertiser. The cost of advertising effort of the advertiser brand \(B_{a}\) determines the advertising efficacy of the advertiser brand. **Hypothesis 1**. : _According to the law of the market, there is usually an upper limit of market saturation for products. When the upper limit is reached, the demand will not increase. This paper assumes that the maximum sales volume of the advertising products is within the maximum saturation of the market._  
**Hypothesis 2**. : _After the commodity appears in the market, the consumers would gradually accept it even though there is no advertisement on the e-commerce platform. According to the law of the market, there naturally exists the growth contribution rate at which the demand of products enhances due to consumers' recognition without brand advertising on the e-commerce platform [13]. Here we set this original growth contribution rate as \(\xi_{0}\). To improve the popularity of products, the merchants are willing to take some measures to promote through brand advertising on the e-commerce platform. Thus, it brings some improvement effects to the sales volume of the advertising products. This is a new derived growth contribution rate which we assume it as \(\xi_{1}\). The total growth contribution rate of consumers' recognition of products through brand advertising channels and non-brand advertising channels on the e-commerce platform is set as \(\xi\) such that \(\xi=\xi_{0}+\xi_{1}\)._  
**Hypothesis 3**. : _The demand function of advertising products is related to the price and total growth contribution rate of products as shown in Equation (1)._  
\[Q(p_{a})=\xi\cdot f(p_{a})=(\xi_{0}+\xi_{1})\cdot f(p_{a}) \tag{1}\]  
**Hypothesis 4**.