Abstract
In this paper, we propose and compare three heterogeneous Cournotian duopolies, in which players adopt best response mechanisms based on different degrees of rationality. The economic setting we assume is described by an isoelastic demand function with constant marginal costs. In particular, we study the effect of the rationality degree on stability and convergence speed to the equilibrium output. We study conditions required to converge to the Nash equilibrium and the possible route to destabilization when such conditions are violated, showing that a more elevated degree of rationality of a single player does not always guarantee an improved stability. We show that the considered duopolies exhibit either a flip or a Neimark–Sacker bifurcation. In particular, in heterogeneous oligopolies models, the Neimark–Sacker bifurcation usually arises in the presence of a player adopting gradient-like decisional mechanisms, and not best response heuristic, as shown in the present case. Moreover, we show that the cost ratio crucially influences not only the size of the stability region, but also the speed of convergence toward the equilibrium.
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Agiza, H.N., Elsadany, A.A.: Nonlinear dynamics in the Cournot duopoly game with heterogeneous players. Phys. A 320, 512–524 (2003)
Agiza, H.N., Elsadany, A.A.: Chaotic dynamics in nonlinear duopoly game with heterogeneous players. Appl. Math. Comput. 149(3), 843–860 (2004)
Agiza, H.N., Hegazi, A.S., Elsadany, A.A.: Complex dynamics and synchronization of a duopoly game with bounded rationality. Math. Comput. Simul. 58(2), 133–146 (2002)
Angelini, N., Dieci, R., Nardini, F.: Bifurcation analysis of a dynamic duopoly model with heterogeneous costs and behavioural rules. Math. Comput. Simul. 79(10), 3179–3196 (2009)
Askar, S.S.: The rise of complex phenomena in Cournot duopoly games due to demand functions without inflection points. Commun. Nonlinear Sci. Numer. Simul. 19(6), 1918–1925 (2014)
Bischi, G.I., Gallegati, M., Naimzada, A.: Symmetry-breaking bifurcations and representative firm in dynamic duopoly games. Ann. Oper. Res. 89, 253–272 (1999)
Bischi, G.I., Kopel, M., Naimzada, A.: On a rent-seeking game described by a non-invertible iterated map with denominator. Nonlinear Anal. 47(8), 5309–5324 (2001)
Bischi, G.I., Naimzada, A.: Global analysis of a dynamic duopoly game with bounded rationality. In: J. Filar, V. Gaitsgory, K. Mizukami (eds.) Advance in Dynamics Games and Applications, 7th International Symposium on Dynamics Games and Applications, pp. 361–385. Shonan Village Ctr (2000)
Bischi, G.I., Naimzada, A., Sbragia, L.: Oligopoly games with local monopolistic approximation. J. Econ. Behav. Organ. 62(3), 371–388 (2007)
Cavalli, F., Naimzada, A.: A cournot duopoly game with heterogeneous players: nonlinear dynamics of the gradient rule versus local monopolistic approach. Appl. Math. Comput. 249, 382–388 (2014)
Cavalli, F., Naimzada, A., Tramontana, F.: Nonlinear dynamics and global analysis of an heterogeneous cournot duopoly with a local monopolistic approach versus a gradient rule with endogenous reactivity. Commun. Nonlinear Sci. Numer. Simul. 23(1–3), 245–262 (2015)
Cournot, A.A.: Researches into the Principles of the Theory of Wealth. Engl. Trans., Irwin Paper Back Classics in Economics (1963)
Den-Haan, W.J.: The importance of the number of different agents in a heterogeneous asset-pricing model. J. Econ. Dyn. Control 25(5), 721–746 (2001)
Dubiel-Teleszynski, T.: Nonlinear dynamics in a heterogeneous duopoly game with adjusting players and diseconomies of scale. Commun. Nonlinear Sci. Numer. Simul. 16(1), 296–308 (2011)
Elsadany, A.A., Tramontana, F.: Heterogeneous triopoly game with isoelastic demand function. Nonlinear Dyn. 68(1–2), 187–193 (2012)
Gao, X., Zhong, W., Mei, S.: Nonlinear cournot oligopoly games with isoelastic demand function: the effects of different behavior rules. Commun. Nonlinear Sci. Numer. Simul. 17(12), 5249–5255 (2012)
Leonard, D., Nishimura, K.: Nonlinear dynamics in the cournot model without full information. Ann. Oper. Res. 89, 165–173 (1999)
Li, T., Ma, J.: The complex dynamics of rc̣ompetition models of three oligarchs with heterogeneous players. Nonlinear Dyn. 74(1–2), 45–54 (2013)
Ma, J., Wu, F.: The application and complexity analysis about a high-dimension discrete dynamical system based on heterogeneous triopoly game with multi-product. Nonlinear Dyn. 77(3), 781–792 (2014)
Naimzada, A., Ricchiuti, G.: Monopoly with local knowledge of demand function. Econ. Model. 28(1–2), 299–307 (2011)
Naimzada, A., Sbragia, L.: Oligopoly games with nonlinear demand and cost functions: two boundedly rational adjustment processes. Chaos Solitons Fractals 29(3), 707–722 (2006)
Naimzada, A., Tramontana, F.: Controlling chaos through local knowledge. Chaos Solitons Fractals 42(4), 2439–2449 (2009)
Poston, T., Stewart, I.: Catastrophe theory and its applications. Pitman Ltd, London (1978)
Puu, T.: Chaos in duopoly pricing. Chaos Solitons Fractals 1(6), 573–581 (1991)
Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, Texts in Applied Mathematics, vol. 37, second edn. Springer, Berlin (2007)
Rand, D.: Exotic phenomena in games and duopoly models. J. Math. Econ. 5(2), 173–184 (1978)
Silvestre, J.: A model of general equilibrium with monopolistic behavior. J. Econ. Theory 16, 425–442 (1977)
Tramontana, F.: Heterogeneous duopoly with isoelastic demand function. Econ. Model. 27(1), 350–357 (2010)
Tuinstra, J.: A price adjustment process in a model of monopolistic competition. Int. Game Theory Rev. 6(3), 417–442 (2004)
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Cavalli, F., Naimzada, A. Nonlinear dynamics and convergence speed of heterogeneous Cournot duopolies involving best response mechanisms with different degrees of rationality. Nonlinear Dyn 81, 967–979 (2015). https://doi.org/10.1007/s11071-015-2044-y
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DOI: https://doi.org/10.1007/s11071-015-2044-y