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 of adjustment 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 of attraction 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.
We study a heterogeneous duopolistic Cournotian game, in which the firms, producing a homogeneous good, have reduced rationality and respectively adopt a "Local Monopolistic Approximation" (LMA) and a gradient-based approach with endogenous reactivity, in an economy characterized by isoelastic demand function and linear total costs. We give conditions on reactivity and marginal costs under which the solution converges to the Cournot-Nash equilibrium. Moreover, we compare the stability regions of the proposed oligopoly to a similar one, in which the LMA firm is replaced by a best response firm, which is more rational than the LMA firm. We show that, depending on costs ratio, the equilibrium can lose its stability in two different ways, through both a flip and a Neimark-Sacker bifurcation. We show that the nonlinear, noninvertible map describing the model can give rise to several coexisting stable attractors (multistability). We analytically investigate the shape of the basins of attractions, in particular proving the existence of regions known in the literature as lobes.
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