Threshold advertising rules in a duopoly framework

In this paper we investigate a duopoly model which describes the dynamics of the firms' advertising decisions and their resulting market shares. The duopolists use threshold advertising rules – increase advertising outlays if the market shares falls below a certain threshold, otherwise decrease them – in order to determine the advertising budgets of the following period. To guarantee that the model is logically consistent, we use a differential-effects version of the well-known MCI model. We show that the resulting time-discrete two-dimensional model can be reduced to a one-dimensional model by applying a suitable transformation. This one-dimensional system contains the same information and we demonstrate by numerical methods that stable, but also periodic and chaotic evolutions of the market shares are possible.