Optimal acquisition and retention strategies in a duopoly model of competition

We propose an optimal customer relationship marketing policy in a duopolistic market, where each firm’s market share depends on the CRM decisions of its own and of its competitors. The evolution process of the number of customers served by each firm is governed by a differential equation in which customer acquisition and retention processes are considered. A differential game model is presented to determine the optimal acquisition and retention expenditures of players. We derive a Nash equilibrium of a duopolistic differential game using the Hamilton–Jacobi–Bellman equations. Our results can be summarized mainly by the following three points. First, the optimal acquisition and retention expenditure strategies depend on each firm’s marginal customer equity, but not on the market share or the number of customers. Second, in response to the variation of the firm’s parameters, if its acquisition effectiveness is greater than its retention effectiveness, the firm would take action by making the same investment decision as its rival’s; instead, if its acquisition effectiveness is lower than its retention effectiveness, its rival would take action by making the different investment decision from the firm’s. Last, besides a mature product market, we also consider a market with remaining potential. In this case, a firm’s marginal customer equity can be decomposed into two components: the marginal customer equity from its own and from its rival’s. We show that the firm’s optimal investments in retention and acquisition are both positively related with a weighted difference between two components of the marginal customer equity.