Closed-loop advertising strategies in a duopoly

Using the Lanchester model to describe the dynamics of the market where two firms compete for customers by advertising, we solve the problem of determining an optimal advertising strategy for maximum discounted profits. We develop both open- and closed-loop strategies and explain the relationship between them. Using a new mathematical approach, we prove that our closed-loop solution is a global Nash equilibrium. The closed-loop strategy is time-variant and depends linearly on the actual market share. The time-variant coefficient incorporates the discount factor; its computation requires the solution of a backward differential equation and a set of two nonlinear differential equations for an initial value problem. The closed-loop advertising expenditures are proportional to the open-loop advertising expenditures and to the square of the competitor's actual market share. This provides a very practical adaptive control rule that allows the manager to adjust the actual advertising expenditure and to deviate from budget. We illustrate the use of our control rule, using data for the period 1968-1984 of the Cola War. Marketing implications of the results are provided.