Feedback advertising strategies in a two-firm differential game: a numerical investigation

We present a procedure for finding feedback strategies numerically in a two-firm differential game (DG). To motivate the discussion, we start with the single-firm optimal control formulation of the problem with a linear revenue term and a nonlinear (quadratic) advertising rate cost term. We find the optimal control to maximise the firm's market share. When two firms compete for the market share, the optimal control model generalises to a differential game formulation with two differential equations (DEs) representing each firm's market share. This formulation extends earlier models with the inclusion of a new competition effect term. We then compute the equilibrium feedback advertising policies of the differential game for each firm as functions of time and the firms' current market shares by solving a system of six DEs. Our analyses are illustrated with several examples including comparative statics and discussions of the managerial implications.