Fallacies of composition in nonlinear marketing models

• We examine some nonlinear dynamic market share attraction models.
• We consider the symmetric case of identical agents and the one-dimensional dynamics of representative agent.
• The properties of the one-dimensional model of representative player may be misleading.
• Asymptotic synchronization of quasi-identical agents may fail.
• Convergence to highly non-symmetric attractors may occur even with very small initial asymmetries.

In this paper we consider some nonlinear discrete-time dynamic models proposed in the literature to represent marketing competition, and we use these models to critically discuss the statement, often made in economic literature, that identical agents behave identically and quasi-identical ones behave in a similar way. We show, through examples and some general mathematical statements, that the one-dimensional model of a representative agent, whose dynamics summarize the common behavior of identical interacting agents, may be misleading. In order to discuss these topics some simple methods for the study of local stability and bifurcations are employed, as well as numerical examples where some results taken from the literature on chaos synchronization are applied to two-dimensional marketing models that exhibit riddling, blowout and other global phenomena related to the existence of measure-theoretic attractors.