Routes to Complexity Induced by Constraints in Cournot Oligopoly Games with Linear Reaction Functions

Within a classical discrete-time Cournot oligopoly model with linear demand and quadratic cost functions, minimum and maximum production constraints are imposed in order to explore their effects on the dynamic of the system. Due to the presence of such constraints, the dynamic model assumes the form of a continuous piecewise linear map of the plane. The study of Nash equilibria of the oligopoly game, together with an analytical and numerical investigation of the different kinds of attractors of the dynamical system, shows how the presence of production constraints generates so called border collision bifurcations, a kind of global bifurcations recently introduced in the literature on non-smooth dynamical systems, which gives rise to a quite rich spectrum of dynamic scenarios, characterized by drastic changes in the qualitative dynamic properties of the system.

[1]  Erik Mosekilde,et al.  Quasiperiodicity and torus breakdown in a power electronic dc/dc converter , 2007, Math. Comput. Simul..

[2]  Laura Gardini,et al.  Border Collision bifurcations in 1D PWL Map with One Discontinuity and Negative Jump: Use of the First Return Map , 2010, Int. J. Bifurc. Chaos.

[3]  Michael Schanz,et al.  On the fully developed bandcount adding scenario , 2008 .

[4]  Franklin M. Fisher,et al.  The Stability of the Cournot Oligopoly Solution: The Effects of Speeds of Adjustment and Increasing Marginal Costs , 1961 .

[5]  Martin Shubik,et al.  Game theory models and methods in political economy , 1977 .

[6]  F. Lamantia.  A Nonlinear Duopoly with Efficient Production-Capacity Levels , 2011 .

[7]  Leon O. Chua,et al.  BIFURCATIONS OF ATTRACTING CYCLES FROM TIME-DELAYED CHUA’S CIRCUIT , 1995 .

[8]  Frank Hahn,et al.  The Stability of the Cournot Oligopoly Solution , 1962 .

[9]  Stephen John Hogan,et al.  Local Analysis of C-bifurcations in n-dimensional piecewise smooth dynamical systems , 1999 .

[10]  James A. Yorke,et al.  Border-collision bifurcations including “period two to period three” for piecewise smooth systems , 1992 .

[11]  Christian Mira,et al.  Chaotic Dynamics in Two-Dimensional Noninvertible Maps , 1996 .

[12]  L. Gardini,et al.  Mathematical properties of a discontinuous Cournot–Stackelberg model , 2011 .

[13]  Volodymyr L. Maistrenko,et al.  On period-adding sequences of attracting cycles in piecewise linear maps , 1998 .

[14]  F. Szidarovszky,et al.  Nonlinear Oligopolies: Stability and Bifurcations , 2009 .

[15]  L. Gardini,et al.  Border collision bifurcations in discontinuous one-dimensional linear-hyperbolic maps , 2011 .

[16]  Michael Schanz,et al.  Multi-parametric bifurcations in a piecewise–linear discontinuous map , 2006 .

[17]  J. Yorke,et al.  Bifurcations in one-dimensional piecewise smooth maps-theory and applications in switching circuits , 2000 .

[18]  Somnath Maity,et al.  Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation. , 2006, Chaos.

[19]  Anna Norin,et al.  Cournot duopoly when the competitors operate under capacity constraints , 2003 .

[20]  J. Geanakoplos,et al.  Multimarket Oligopoly: Strategic Substitutes and Complements , 1985, Journal of Political Economy.

[21]  Michael Schanz,et al.  On multi-parametric bifurcations in a scalar piecewise-linear map , 2006 .

[22]  Mario di Bernardo,et al.  Qualitative theory of non-smooth dynamical systems , 2008 .

[23]  Laura Gardini,et al.  Global bifurcations in duopoly when the Cournot Point is Destabilized via a Subcritical Neimark bifurcation , 2006, IGTR.

[24]  Mario di Bernardo,et al.  Piecewise smooth dynamical systems , 2008, Scholarpedia.

[25]  Michael Kopel,et al.  Simple and complex adjustment dynamics in Cournot duopoly models , 1996 .

[26]  Gian Italo Bischi,et al.  Nonlinear duopoly games with positive cost externalities due to spillover effects , 2002 .

[27]  R. D. Theocharis.  On the Stability of the Cournot Solution on the Oligopoly Problem , 1960 .

[28]  Gian Italo Bischi,et al.  Equilibrium selection in a nonlinear duopoly game with adaptive expectations , 2001 .

[29]  Arjen van Witteloostuijn,et al.  CHAOTIC PATTERNS IN COURNOT COMPETITION , 1990 .

[30]  Celso Grebogi,et al.  Border collision bifurcations in two-dimensional piecewise smooth maps , 1998, chao-dyn/9808016.

[31]  Luigi Montrucchio,et al.  Dynamic complexity in duopoly games , 1986 .

[32]  Tönu Puu,et al.  Chaos in duopoly pricing , 1991 .

[33]  Topological Entropy of Cournot-Puu Duopoly , 2010 .

[34]  Laura Gardini,et al.  Multistability and cyclic attractors in duopoly games , 2000 .

[35]  Richard E. Quandt,et al.  Comments on the Stability of the Cournot Oligipoly Model , 1961 .

[36]  S. Rassenti,et al.  The stability of the cournot solution under adaptive expectation: A further generalization , 1994 .

[37]  P. Holmes.  Chaotic Dynamics , 1985, IEEE Power Engineering Review.

[38]  Leon O. Chua,et al.  Cycles of Chaotic Intervals in a Time-delayed Chua's Circuit , 1993, Chua's Circuit.

[39]  J. Bertrand.  Cournot oligopoly: Review of Walras's Théorie mathématique de la richesse sociale and Cournot's Recherches sur les principes mathématiques de la théorie des richesses , 1989 .

[40]  R. Battalio,et al.  Selection Dynamics, Asymptotic Stability, and Adaptive Behavior , 1994, Journal of Political Economy.

[41]  David A. Rand,et al.  Exotic phenomena in games and duopoly models , 1978 .

[42]  Koji Okuguchi,et al.  The Stability of The Cournot Oligopoly Solution: A Further Generalization , 1964 .

[43]  Michael Schanz,et al.  Border-Collision bifurcations in 1D Piecewise-Linear Maps and Leonov's Approach , 2010, Int. J. Bifurc. Chaos.

[44]  Erik Mosekilde,et al.  Bifurcations and chaos in piecewise-smooth dynamical systems , 2003 .