Overtaking optimality in a discrete-time advertising game

In this paper, advertising competition among $ m $ firms is studied in a discrete-time dynamic game framework. Firms maximize the present value of their profits which depends on their advertising strategy and their market share. The evolution of market shares is determined by the firms' advertising activities. By employing the concept of the discrete-time potential games of Gonzalez-Sanchez and Hernandez-Lerma (2013), we derived an explicit formula for the Nash equilibrium (NE) of the game and obtained conditions for which the NE is an overtaking optimal. Moreover, we analyze the asymptotic behavior of the overtaking NE where the convergence towards a unique steady state (turnpike) is established.