Abstract: In the context of the digital economy, the rapid development of the live-streaming industry has significantly promoted social and economic progress. By 2023, the number of short video users in China had reached 1.012 billion, and the number of live-streaming users had stood at 751 million, making the live-streaming industry the leader in China’s social media sector. This surge in internet celebrity-driven live commerce has posed a considerable challenge to traditional direct-to-consumer e-commerce. Consequently, brand manufacturers face a crucial dilemma: how to leverage internet celebrity live-streaming channels to maximize profits, considering the impact of internet celebrities and the chosen mode of cooperation?
To explore the optimal cooperation model between internet celebrities and brand manufacturers on live-streaming platforms, while accounting for the continuous nature of live-stream traffic and time, this paper constructs a differential game system involving one internet celebrity and brand manufacturers selling similar products. By employing the Hamilton-Jacobi-Bellman equation, we analyze the optimal decisions regarding effort levels and corresponding payoffs for both internet celebrities and brand manufacturers under three cooperation models: the decentralized decision model, the centralized decision model and the fairness concern model. Through theoretical analysis of the effort levels and overall system revenue under these three cooperation models, coupled with the numerical experiment using real data from the live e-commerce analytics platform “Douchacha,” we derive the following conclusions.
(1)In the centralized decision model, both the effort levels of the internet celebrity and the brand manufacturer, as well as the level of live-streaming traffic, are maximized. This model facilitates the highest degree of cooperation and synergy between the parties, thereby significantly enhancing market promotion effectiveness and overall revenue.
(2)In the decentralized decision model, each party makes independent decisions without global cooperation and coordination, which limits the efforts of both the internet celebrities and the brand manufacturers. Consequently, the system’s overall performance is the lowest in this model due to the lack of a cohesive strategy and mutual alignment.
(3)In the fairness concern model, the system achieves the highest overall profit. By appropriately reducing the fairness concern of the internet celebrity towards each brand manufacturer or increasing the commission paid by the brand manufacturers to the internet celebrity, both the internet celebrity’s and the brand manufacturers’ profits can be effectively increased. Such coordination can ensure the maximization of the interests of all parties involved, thereby achieving a true win-win situation.
Through a comparative analysis of theoretical and numerical experiments, it is evident that the centralized decision model and the fairness concern model facilitate deeper collaboration between internet celebrities and brand manufacturers more effectively than the decentralized decision model. This approach not only yields higher returns for both parties but also supports the healthy and efficient development of the internet celebrity economy, fostering long-term growth in the industry.

Key words: internet celebrity, brand merchants, differential game, optimal strategy, fairness concern

摘要： 在数字经济的背景下,直播行业的快速发展对社会经济进步起到了重要的推动作用。本文聚焦于直播平台中网红主播与品牌商之间的合作模式,同时考虑网红主播的引流水平以及时间连续性,构建了由1个网红主播和n个同质品牌商组成的微分博弈系统;运用哈密尔顿—雅可比—贝尔曼方程,分析在分散式决策模型、集中式决策模型和公平关注模型下网红主播和品牌商关于各自努力程度的最优决策以及相应的最优收益。通过对以上三种合作模型下的理论分析与数值实验对比,我们发现:在集中式完全合作模型下,网红主播的引流水平最高。在分散式决策模型下,各方独立决策,缺乏全局性的合作与协调,导致网红主播和品牌商的努力程度受限,进而使得系统整体收益在此模型下表现最低。在公平关注模型下,系统整体收益最高,适当降低网红主播对各品牌商的公平关注或提高品牌商支付网红主播佣金,可以有效促进网红主播与品牌商双方利润的增长。

关键词: 网红主播, 品牌商, 微分博弈, 最优策略, 公平关注