An explicit Nash equilibrium for a market share attraction game

In competitive marketing, the speed of generating the best price has become as critical as its reliability. In this study, we aim to design a practical marketing management tool. We consider a non-cooperative marketing environment with multiple substitute products, where total market size is moderately price-sensitive. The price-demand relations are determined by a market share attraction model, where the attraction of each product is a linear function of its price. The product's brand image is reflected in the parameters of this linear function. For the general case of multiple substitute products, we derive explicit expressions for the best-response functions. For the specific case of two substitute products, we derive closed form expressions for the prices at Nash equilibrium. These expressions help managers in changing their marketing instruments other than price, so as to obtain substantial individual profits. We show how our closed form Nash equilibrium enables the examination of the profit loss due to competition. Relevant for practice is the fact that our model can be easily calibrated. We provide a simple procedure for estimating the model parameters.