Competition for Market in the Last Stage of Product Life-cycle: a Differential Games Approach

ABSTRACT. After stable sale in the second stage of product life cycle, the sale will decrease gradually. This paper deals with competition in the last stage of product life cycle. A differential game model will be set up for competition in this stage. We base the dynamics on Lanchester type competition and natural decrease of sales. Because the length of product life will be affected by the advertising, the n competitors’ controls/advertising policies will be coupled in the coefficients of natural decrease. Infinite time horizon will be involved. Besides open-loop and closed-loop controls, we will discuss another type of competition, in which the roles of competitors are not symmetric: some competitors adopt open-loop controls and others adopt closed-loop controls. From Pontryagin optimal condition we derive Two-Point-Boundary-Value Problems(TPBVP). We will solve the TPBVP by an algorithm based on Newton’s method. In the algorithm we will use random perturbation technique to generate Jacobian matrix and prevent the Jacobian matrix from being singular. From numerical results we draw general rules about competition policies in the final stage. From the numerical results we will also see that in some markets, small companies following the control/advertising policies of bigger companies is a good strategy.