Global Analysis of a Dynamic Duopoly Game with Bounded Rationality

A dynamic Cournot duopoly game, characterized by finns with bounded rationality, is represented by a discrete-time dynamical system of the plane. Conditions ensuring the local stability ofa Nash equilibrium, under a local (or myopic) adjustment process, are given, and the influence ofmarginal costs and speeds ofadjustment of the two finns on stability is studied. The stability loss ofthe Nash equilibrium, as some parameter ofthe model is varied, gives rise to more complex (periodic or chaotic) attractors. The main result of this paper is given by the exact detennination ofthe basin ofattraction ofthe locally stable Nash equilibrium (or other more complex bounded attractors around it), and the study of the global bifurcations that change the structure ofthe basin from a simple to a very complex one, with consequent loss ofpredictability, 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.