Analysis of a Dynamical Cournot Duopoly Game with Distributed Time Delay

  • Nicoleta SÎRGHI West University of Timişoara
  • Mihaela NEAMȚU West University of Timișoara
Keywords: Cournot duopoly game, Cournot Model Bounded Rationality, distributed time delay, local stability, Hopf bifurcation


The aim of the paper is to analyze the dynamic model of the Cournot duopoly game with bounded rationality associated to two firms.  We consider the cost function of the first firm as nonlinear and for the second firm as linear. The players do not have a complete knowledge of the market and they follow a bounded rationality adjustment process based on the estimation of the marginal profit. Also, the distributed time delay is introduced, because the decisions at the current time depend on the average past decisions.  The mathematical model is described by a distributed delay differential system with two nonlinear equations. The study for the local stability of the Nash equilibrium point is carried out in the case of two types of kernels: weak (exponential) and Dirac. A change in local stability of the equilibrium point, from stable to unstable, implies a Hopf bifurcation. The delays are considered as bifurcation parameters. In some conditions of the parameters of the model, we have proved that a family of periodic solutions bifurcates from the equilibrium point when the bifurcation parameter passes through a critical value. Numerical simulations are performed to illustrate the effectiveness of our results. Finally, conclusions and future researches are provided.


Agiza, H. N. (1999). On the Analysis of Stability, Bifurcation, Chaos and Chaos Control of Kopel Map. Chaos, Solitons & Fractals 01/1999. doi:10.1016/S0960-0779(98)00210-0

Agiza, H. N., & Elsadany, A. A. (2003). Nonlinear dynamics in the Cournot duopoly game with heterogeneous players. Physica A, 320(1–4), 512–524.

Agiza, H. N., & Elsadany, A. A. (2004). Chaotic dynamics in nonlinear duopoly game with heterogeneous players. Applied Mathematics and Computation, 149(3), 843–860.

Agiza, H. N., Elsadany, A. A., & El-Dessoky, M. M. (2013). On a New Cournot Duopoly Game. Journal of Chaos, 2013, 5 pages.

Agiza, H. N., Hegazi, A. S., & Elsadany, A. A. (2001). The dynamics of Bowley’smodel with bounded rationality. Chaos, Solitons and Fractals, 12(9), 1705–1717.

Agiza, H. N., Hegazi, A. S., & Elsadany, A. A. (2002). Complex dynamics and synchronization of a duopoly game with bounded rationality. Mathematics and Computers in Simulation, 58(2), 133–146.

Binmore, B. (1999). Jeux et theorie des jeux, Bruxelles: Ed. De Boeck Universite.

Chiarella, C., & Khomin, A. (1996). An analysis of the complex dynamic behavior of the nonlinear oligopoly models with time lags. Chaos, Solitons and Fractals,7(12), 2049–2065.

Dubiel-Teleszynski, T. (2011). Nonlinear dynamics in a heterogeneous duopoly game with adjusting players and diseconomies of scale. Communications in Nonlinear Science and Numerical Simulation, 16(1), 296–308.

Elsadany, A. A. (2010). Dynamics of a delayed duopoly game with bounded rationality. Mathematical and Computer Modelling, 52(9-10), 1479–1489.

Fan, Y. Q., Xie, T., & Du, J. G. (2012). Complex dynamics of duopoly game with heterogeneous players: a further analysis of the output model. Applied Mathematics and Computation, 218(15), 7829–7838.

Hassard, B. D., Kazarinoff, N. D., & Wan, Y. H. (1981). Theory and Applications of Hopf Bifurcation. London Mathematical Society Lecture Note Series, vol. 41. Cambridge, UK: Cambridge University Press.

Kopel, M. (1996). Simple and complex adjustment dynamics in Cournot Duopoly model. Chaos, Solitons and Fractals, 7(12), 2031–2048.

Ma, J. H., & Ji, W. Z. (2009). Complexity of repeated game model in electric power triopoly. Chaos, Solitons and Fractal, 40(4), 1735–1740.

Ma, J. H., & Tu, H. (2012). Complexity of a Duopoly Game in the Electricity Market with Delayed Bounded Rationality. Discrete Dynamics in Nature and Society, 2012, Article ID 698270. doi:10.1155/2012/698270.

Mircea, G., Neamţu, M., & Opriş, D. (2004). Bifurcaţii Hopf pentru sisteme dinamice cu argument întârziat şi aplicaţii. Timisoara, Romania: Mirton.

Neamtu, M. (2010). Deterministic and stochastic Cournot duopoly games with tax evasion. WSEAS Transactions on Mathematics, 9(8), 618–627.

Neamtu, M., Sîrghi, N., Babaita, C., & Antonie-Nitu, R. (2011). Discrete-time deterministic and stochastic triopoly game with heterogeneous players and delay. International Journal of Mathematical models and methods in applied sciences, 5(2), 343–350.

Onozaki, T., Sieg, G., & Yokoo, M. (2003). Stability, chaos and multiple attractors: a single agentmakes a difference. Journal of Economic Dynamics and Control, 27(10), 1917–1938.

Sheng, Z., Du, J., Mei Q., & Huang, T. (2013). New Analyses of Duopoly Game with Output Lower Limiters. Abstract and Applied Analysis, 2013.

Sirghi, N. (2008). Microeconomics Advanced. Theory and applications (in Romanian) Timisoara, Romania: Mirton.

Yassen, M. T., & Agiza, H. N. (2003). Analysis of a duopoly game with delayed bounded rationality. Applied Mathematics and Computation, 138(2-3), 387–402.

Zhang, J. X., Da, Q. L., & Wang, Y. H. (2007). Analysis of nonlinear duopoly game with heterogeneous players. Economic Modelling, 24(1),138–148.

How to Cite
SÎRGHI, N., & NEAMȚU, M. (2015). Analysis of a Dynamical Cournot Duopoly Game with Distributed Time Delay. Timisoara Journal of Economics and Business, 8(1s), 1-13. Retrieved from