Modeling Market Share Dynamics Under Advertising Effort and Word-of-Mouth Interactions Between Customers

This paper presents a complete analysis of a new model of market share dynamics under advertising effort, taking into account word-of-mouth (WOM) interactions between satisfied customers, dissatisfied defectors, as well as undecided customers. The populations of satisfied customers and defectors are both modeled as competing predators preying on a population of undecided customers, using Lotka-Volterra-type interaction terms, which have also been used in a related, but different, class of WOM and electronic WOM models. The proposed model describes the dynamics of market share from arbitrary nonnegative initial conditions, up to and including the market in which there are no longer any undecided customers, thereby extending both the classical Vidale-Wolfe model in which the market with no undecided customers is never reached at equilibrium due to the positive decay term, as well as the Lanchester model which only deals with the market with no undecided customers. Another new feature of the proposed model is that, even under constant advertising effort, and fixed values of interaction coefficients, different outcomes can arise, depending on the initial fractions of satisfied customers and defectors. The design of a class of advertising policies that attain a desired market share is also presented, with corresponding numerical simulations.