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Models of sales response to advertising, also referred to as market share dynamics under advertising have been the concern first of economists and later of researchers in marketing, starting from the late 1950s. Vidale and Wolfe, in 1957, proposed their now eponymous model based on sales data. Since then, most of the work in the field has centered around the application of optimal control methods, leading to open-loop control of the Vidale-Wolfe and variants, in order to achieve a given market share, while minimizing advertising expenditure. The extension to duopolies was made by Deal, in the continuous-time case, and is denominated the Vidale-Wolfe-Deal (VWD) model. In discrete-time VWD models and variants, two firms compete for market share, in a dynamic game setting , described by a pair of difference equations. This paper studies these dynamic games, using the natural concept of one step ahead optimal control, in which each firm optimizes its own performance index at the next step, and only has access to some information about its competitor. Two cases are studied: with and without stipulating target market shares for each firm. It is shown that when target market shares are not specified , for the VWD model, limit cycles of large period can occur when each firm uses linear performance indices, while multiple equilibria may arise when quadratic performance indices are used. Three other proposed models result in games that lead to equilibria and do not have limit cycle behavior. When target market shares are specified, convergence to an equilibrium occurs for all the models proposed in this paper. |
