Dynamic Budget Allocation in Competitive Search Advertising: A Differential Game Approach

Abstract

Over the past decade, sponsored search advertising has evolved into a major advertising outlet on the Internet. Given the competitive and dynamic nature of the search advertising market, it has long been a great challenge for advertisers to identify their optimal advertising strategies. Using a differential game framework, we propose a dynamic budget allocation model to explore the optimal advertising strategies for advertisers engaged in a duopolistic competition. The proposed model captures the dynamic nature of the search advertising market by incorporating two important advertising variables, namely, the time-varying quality score and the potential market revenue of advertisers, which are specific to the unique context of sponsored search advertising. In this model, we first identify and examine the steady-state Nash equilibrium strategies of the advertisers, and then investigate the optimal paths for the open-loop Nash equilibrium. In addition, we provide an algorithm to compute the optimal path of budget and the corresponding trajectory of market share for the advertisers, and conduct computational experiments to validate our analytical findings. One important finding in this study is that the effects of the two advertising variables on the equilibrium budget levels are non-monotonic. Moreover, our results indicate that as an advertiser improves her quality scores, her market share in equilibrium will increase accordingly; in the meantime, however, the advertiser may need to escalate her advertising expenditure. This finding to some extent challenges the commonly-held view that higher quality scores lead to lower optimal advertising expenditures. Another interesting finding suggests that advertisers should set a budget cap even in a competitive environment to avoid the potential risk of over-spending in sponsored search advertising.