Modelling a dynamic market potential: A class of automata networks for diffusion of innovations

Abstract

Innovation diffusion processes are generally described at aggregate level with models like the Bass Model (BM) and the Generalized Bass Model (GBM). However, the recognized importance of communication channels between agents has recently suggested the use of agent-based models, like Cellular Automata. We argue that an adoption or purchase process is nested in a communication network that evolves dynamically and indirectly generates a latent non-constant market potential affecting the adoption phase. Using Cellular Automata we propose a two-stage model of an innovation diffusion process. First we describe a communication network, an Automata Network, necessary for the “awareness” of an innovation. Then, we model a nested process depicting the proper purchase dynamics. Through a mean field approximation we propose a continuous representation of the discrete time equations derived by our nested two-stage model. This constitutes a special non-autonomous Riccati equation, not yet described in well-known international catalogues. The main results refer to the closed form solution that includes a general dynamic market potential and to the corresponding statistical analysis for identification and inference. We discuss an application to the diffusion of a new pharmaceutical drug.