## 1 Introduction
However, the above-mentioned literature did not consider the impact of manufacturer's advertising on product market demand and did not take the relationship between manufacturer's advertising share rate and retailer's advertising into account. They only conducted a single-cycle game analysis based on the short-term impact of advertising on product sales. Neither the time dynamics nor the long-term effects were considered. [PERSON] and [PERSON] et al. [6] built the optimal advertising strategic model and analyzed the change of market power structure contrast between manufacturers and retailers based on their peer-peer relationship with the typical cases of the Wal-Mart and P&G. [PERSON] and [PERSON] et al. [7] modified [PERSON] & [PERSON]'s model, and put forward a two-layer supply chain advertising cooperation problem in which the manufacturers directly offered discounts to the consumers in combination with the elasticity of demand in the market environment. [PERSON] and [PERSON] [8] presented the game problem of cooperative manufacturers and retailers on one stage, and further analyzed the multi-stage game equilibriums of both channel members. Based on the study of the retailers' advertising behavior influence on brand image, [PERSON] _et al._[9; 10] analyzed the optimal advertising strategy and the transaction efficiency of both parties in the supply chain. These above literatures considered the influence of local advertising of retailers and national advertising of manufacturers on sales volume, but there were many disputes on the correlation between advertising expenditure of manufacturers and retailers. Most of the above literatures carried out the study on the advertising cooperation arrangement and advertising cost sharing in a dynamic environment to prove that advertising cooperation is an effective means to improve marketing channel profits. However, little literature has studied the advertising cooperation strategies of both sides based on the deep analysis of the influence of advertising cost on brand effect, and the modes of the advertising cooperation. At present, the application of closed-loop supply chain in enterprises is increasing and the market demand as well. It has become a new development trend of logistics and supply chain management. Many scholars have conducted deep research on closed-loop supply chain. [PERSON] and [PERSON] [11] discussed a two-stage closed-loop supply chain game in which the remanufacturer allocated the residual value of the returned item and determined whether to manage the collection of end-of-use products exclusively or outsource it to a retailer or a third-party service provider. They found that the manufacturers tended to outsource the production collection when the outsourcing performedenvironmentally and operationally better and the manufacturers were generally more sensitive to environmental performance than operational performance. [PERSON] et al. (2012) made a study of the closed-loop supply chain under four reverse channel structures, in which the central planner, the manufacturer, the retailer or the third party acted as collectors of second-hand products respectively, and the demand depended on the retailer's marketing efforts. They first achieved supply chain profitability in the centralized and decentralized supply chain management service center, and provided the optimal marketing effort, collection rate and pricing decision for supply chain members. In addition, then they considered the distributional fairness of the retailer and the potential recycle cost advantages by the retailer and the third party as the two directions to extend the manufacturer collection model. The above researches have attracted many people's attention. Likewise, the cooperation between e-commerce platforms and advertisers proposed in this paper is similar to the game in the supply chain. The e-commerce platform and the advertiser play the different role in the market power structure in such advertising cooperation.