تخفیف قیمت گذاری در بازاریابی شفاهی: یک رویکرد کنترل بهینه Discount pricing in word-of-mouth marketing: An optimal control approach

1. Introduction Dynamic pricing, which is defined as the dynamic adjustment of market price, has been widely adopted by marketing researchers and practitioners [1–6]; this is because price is one of the most controllable variables. In the past, dynamic pricing strategies were widely applied to traditional marketing, with a significant cost associated with changing prices [7–9]. With the advent of the Internet marketing, the overhead has greatly diminished [10–13]. Dynamic discount pricing is a common form of dynamic pricing. In contrast to a merely low price, a discounted price typically implies the high quality and unusual bargain of the item and, hence, has strong attraction to rational customers. Consequently, the design and evaluation of dynamic discount pricing strategies have long been a research hotspot in the marketing field [14–20]. As compared to traditional advertizing marketing, the emerging word-of-mouth (WOM) marketing can bring about higher marketing profit at a much lower cost [21–25]; this is especially the case with the popularity of online social networks [26]. To evaluate the performance of different WOM marketing strategies, a number of dynamic models capturing WOM spreading processes have been suggested [27–36]. However, these dynamic models are all population-level, which cannot accommodate the WOM network. As a result, the accuracy of the estimated performance is questionable. To overcome this flaw, individual-level dynamic models, which have been widely applied to areas such as epidemic spreading [37–40], malware spreading [41–47] and cybersecurity [48–50], should be borrowed to more accurately characterize WOM spreading processes. The optimal control theory provides an appropriate mathematical framework for the design of better dynamic strategies [51, 52]. For instance, the optimal control approach has been applied to the dynamic control of malware [53–55], where the virus spreading models accommodate the virus propagation network. To the best of our knowledge, to date the dynamic pricing in WOM marketing has not been dealt with in the framework of optimal control theory. This paper addresses the dynamic discount pricing in WOM marketing by the optimal control approach. First, an individual-level differential dynamic model capturing WOM spreading processes is established. This model accurately characterizes the dynamic discount mechanism and perfectly accommodates the WOM network. On this basis, the expected profit of a WOM marketing champaign is evaluated, and the problem of finding an optimal discount strategy boils down to an optimal control problem.