A Stackelberg viral marketing design for two competing players

A Stackelberg duopoly model in which two firms compete to maximize their market share is considered. The firms offer a service/product to customers that are spread over several geographical regions (e.g., countries, provinces, or states). Each region has its own characteristics (spreading and recovery rates) of each service propagation. We consider that the spreading rate can be controlled by each firm and is subject to some investment that the firm does in each region. One of the main objectives of this work is to characterize the advertising budget allocation strategy for each firm across regions to maximize its market share when competing. To achieve this goal we propose a Stackelberg game model that is relatively simple while capturing the main effects of the competition for market share. {By characterizing the strong/weak Stackelberg equilibria of the game, we provide the associated budget allocation strategy.} In this setting, it is established under which conditions the solution of the game is the so-called ``winner takes all". Numerical results expand upon our theoretical findings and we provide the equilibrium characterization for an example.