Dynamic Advertising Games in Duopolies Under One-Step-Ahead Optimal Control

Abstract

In discrete-time Vidale–Wolfe–Deal duopoly 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, under sequential and parallel game playing procedures. 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.