Global Analysis of a Dynamic Duopoly Game with Bounded Rationality

A dynamic Cournot duopoly game, characterized by firms with bounded rationality, is represented by a discrete-time dynamical system of the plane. Conditions ensuring the local stability of a Nash equilibrium, under a local (or myopic) adjustment process, are given, and the influence of marginal costs and speeds of adjustment of the two firms on stability is studied. The stability loss of the Nash equilibrium, as some parameter of the model is varied, gives rise to more complex (periodic or chaotic) attractors. The main result of this paper is given by the exact determination of the basin of attraction of the locally stable Nash equilibrium (or other more complex bounded attractors around it), and the study of the global bifurcations that change the structure of the basin from a simple to a very complex one, with consequent loss of predictability, as some parameters of the model are allowed to vary. These bifurcations are studied by the use of critical curves, a relatively new and powerful method for the study of noninvertible two-dimensional maps.