November  2021, 26(11): 5905-5923. doi: 10.3934/dcdsb.2021069

Stability switching and its directions in cournot duopoly game with three delays

1. 

Department of Economics, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo, 192-0393, Japan

2. 

Department of Mathematics, Corvinus University, Budapest, Fövám tér 8, 1093, Hungary

 

Received  September 2020 Revised  November 2020 Published  November 2021 Early access  March 2021

Fund Project: The authors appreciate the constructive suggestions and comments by anonymous referees which have greatly helped to improve the paper. The first author highly acknowledges the financial supports from the Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Research (C) 20K01566). The usual disclaimers apply

A three-delay duopoly is considered where the firms have identical implementation delays with different information delays. The equilibrium is locally asymptotically stable without delays however this stability is lost with increasing values of the delays. The stability properties of the equilibrium depend on the common implementation delay of the firms and on the sum of the two information delays. The stability switching curves are first analytically characterized and illustrated, and then the direction of the stability switching is determined at each point of the curves. The possibility of multiple pure imaginary eigenvalues is also discussed when the directions of the stability switches cannot be determined. Simulation examples illustrate the theoretical results.

Citation: Akio Matsumoto, Ferenc Szidarovszky. Stability switching and its directions in cournot duopoly game with three delays. Discrete and Continuous Dynamical Systems - B, 2021, 26 (11) : 5905-5923. doi: 10.3934/dcdsb.2021069
References:
[1] R. Bellman and K. L. Cooke, Difference-Differential Equations, Academic Press, New York, 1963. 
[2]

G. I. Bischi, C. Chiarella, M. Kopel and F. Szidarovszky, Nonlinear Oligopolies: Stability and Bifurcations, Springer, Heidelberg, 2010. doi: 10.1007/978-3-642-02106-0.

[3]

K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, Journal of Mathematical Analysis and Applications, 86 (1982), 592-627.  doi: 10.1016/0022-247X(82)90243-8.

[4]

A. Cournot, Recherches sur les Principes Mathématiques de la Théorie des Richessess, Researches into the Mathematical Principles of Theory of Wealth, Hachette, Paris, 1833, Kelley, New York, 1960.

[5] J. W. Friedman, Game Theory with Applications to Economics, Oxford University Press, New York, Oxford, 1986. 
[6]

L. GoriL. Guerrini and M. Sodini, A continuous time Cournot duopoly with delays, Chaos, Solitons and Fractals, 79 (2015), 166-177.  doi: 10.1016/j.chaos.2015.01.020.

[7]

K. GuS. Niculescu and J. Chen, On stability crossing curves for general systems with two delays, Journal of Mathematical Analysis and Applications, 311 (2005), 231-252.  doi: 10.1016/j.jmaa.2005.02.034.

[8]

L. Guerrini, A. Matsumoto and F. Szidarovszky, Delay Cournot duopoly models revisited, Chaos: An Interdisciplinary Journal of Nonlinear Science, 28 (2018), 093113, 10 pp. doi: 10.1063/1.5020903.

[9]

F. Hahn, The stability of the Cournot oligopoly solution, Review of Economic Studies, 29 (1962), 329-331. 

[10]

T. D. Howroyd and A. M. Russel, Cournot oligopoly models with time delays, Journal of Mathematical Economics, 13 (1984), 97-103.  doi: 10.1016/0304-4068(84)90009-0.

[11]

X. Lin and H. Wang, Stability analysis of delay differential equations with two discrete delays, Canadian Applied Mathematics Quarterly, 20 (2012), 519-533. 

[12]

A. Matsumoto and F. Szidarovszky, Dynamic Oligopolies with Time Delays, Springer Nature, Singapore, 2018. doi: 10.1007/978-981-13-1786-6.

[13]

A. Matsumoto and F. Szidarovszky, Nonlinear Cournot duopoly with implementation delays, Chaos, Solitons and Fractals, 79 (2015), 157-165.  doi: 10.1016/j.chaos.2015.05.010.

[14]

A. MatsumotoF. Szidarovszky and H. Yosida, Dynamics in linear Cournot duopolies with two time delays, Computational Economics, 38 (2011), 311-327. 

[15]

M. McManus and R. Quandt, Comments on the stability of the Cournot oligopoly model, Review of Economic Studies, 28 (1964), 136-139. 

[16]

H. Nikaidô and K. Isoda, Note on noncooperative convex games, Pacific Journal of Mathematics, 5 (1955), 807-815.  doi: 10.2140/pjm.1955.5.807.

[17]

K. Okuguchi, Expectations and Stability in Oligopoly Models, Springer, Berlin, 1976.

[18]

K. Okuguchi and F. Szidarovszky, The Theory of Oligopoly with Multi-product Firms, 2nd. ed, Springer, Berlin, 1999. doi: 10.1007/978-3-662-02622-9.

[19]

J. B. Rosen, Existence and uniqueness of equilibrium points for concave $n$-person games, Econometrica, 33 (1965), 520-534.  doi: 10.2307/1911749.

[20]

Y. SongM. Han and Y. Peng, Stability and Hopf bifurcation in a competitive Lotka-Volterra system with two delays, Chaos, Solitions and Fractals, 22 (2004), 1139-1148.  doi: 10.1016/j.chaos.2004.03.026.

[21]

F. Szidarovszky, On the oligopoly game, DM-70-1, Karl Marx University of Economics, Budapest, Hungary, 1970.

[22]

R. Theocharis, On the stability of the Cournot solution on the oligopoly problem, Review of Economic Studies, 27 (1960), 133-134. 

[23]

J. ZhangZ. JinJ. Yan and G. Sun, Stability and Hopf bifurcation in a delayed competition system, Nonlinear Analysis, 70 (2009), 658-670.  doi: 10.1016/j.na.2008.01.002.

show all references

References:
[1] R. Bellman and K. L. Cooke, Difference-Differential Equations, Academic Press, New York, 1963. 
[2]

G. I. Bischi, C. Chiarella, M. Kopel and F. Szidarovszky, Nonlinear Oligopolies: Stability and Bifurcations, Springer, Heidelberg, 2010. doi: 10.1007/978-3-642-02106-0.

[3]

K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, Journal of Mathematical Analysis and Applications, 86 (1982), 592-627.  doi: 10.1016/0022-247X(82)90243-8.

[4]

A. Cournot, Recherches sur les Principes Mathématiques de la Théorie des Richessess, Researches into the Mathematical Principles of Theory of Wealth, Hachette, Paris, 1833, Kelley, New York, 1960.

[5] J. W. Friedman, Game Theory with Applications to Economics, Oxford University Press, New York, Oxford, 1986. 
[6]

L. GoriL. Guerrini and M. Sodini, A continuous time Cournot duopoly with delays, Chaos, Solitons and Fractals, 79 (2015), 166-177.  doi: 10.1016/j.chaos.2015.01.020.

[7]

K. GuS. Niculescu and J. Chen, On stability crossing curves for general systems with two delays, Journal of Mathematical Analysis and Applications, 311 (2005), 231-252.  doi: 10.1016/j.jmaa.2005.02.034.

[8]

L. Guerrini, A. Matsumoto and F. Szidarovszky, Delay Cournot duopoly models revisited, Chaos: An Interdisciplinary Journal of Nonlinear Science, 28 (2018), 093113, 10 pp. doi: 10.1063/1.5020903.

[9]

F. Hahn, The stability of the Cournot oligopoly solution, Review of Economic Studies, 29 (1962), 329-331. 

[10]

T. D. Howroyd and A. M. Russel, Cournot oligopoly models with time delays, Journal of Mathematical Economics, 13 (1984), 97-103.  doi: 10.1016/0304-4068(84)90009-0.

[11]

X. Lin and H. Wang, Stability analysis of delay differential equations with two discrete delays, Canadian Applied Mathematics Quarterly, 20 (2012), 519-533. 

[12]

A. Matsumoto and F. Szidarovszky, Dynamic Oligopolies with Time Delays, Springer Nature, Singapore, 2018. doi: 10.1007/978-981-13-1786-6.

[13]

A. Matsumoto and F. Szidarovszky, Nonlinear Cournot duopoly with implementation delays, Chaos, Solitons and Fractals, 79 (2015), 157-165.  doi: 10.1016/j.chaos.2015.05.010.

[14]

A. MatsumotoF. Szidarovszky and H. Yosida, Dynamics in linear Cournot duopolies with two time delays, Computational Economics, 38 (2011), 311-327. 

[15]

M. McManus and R. Quandt, Comments on the stability of the Cournot oligopoly model, Review of Economic Studies, 28 (1964), 136-139. 

[16]

H. Nikaidô and K. Isoda, Note on noncooperative convex games, Pacific Journal of Mathematics, 5 (1955), 807-815.  doi: 10.2140/pjm.1955.5.807.

[17]

K. Okuguchi, Expectations and Stability in Oligopoly Models, Springer, Berlin, 1976.

[18]

K. Okuguchi and F. Szidarovszky, The Theory of Oligopoly with Multi-product Firms, 2nd. ed, Springer, Berlin, 1999. doi: 10.1007/978-3-662-02622-9.

[19]

J. B. Rosen, Existence and uniqueness of equilibrium points for concave $n$-person games, Econometrica, 33 (1965), 520-534.  doi: 10.2307/1911749.

[20]

Y. SongM. Han and Y. Peng, Stability and Hopf bifurcation in a competitive Lotka-Volterra system with two delays, Chaos, Solitions and Fractals, 22 (2004), 1139-1148.  doi: 10.1016/j.chaos.2004.03.026.

[21]

F. Szidarovszky, On the oligopoly game, DM-70-1, Karl Marx University of Economics, Budapest, Hungary, 1970.

[22]

R. Theocharis, On the stability of the Cournot solution on the oligopoly problem, Review of Economic Studies, 27 (1960), 133-134. 

[23]

J. ZhangZ. JinJ. Yan and G. Sun, Stability and Hopf bifurcation in a delayed competition system, Nonlinear Analysis, 70 (2009), 658-670.  doi: 10.1016/j.na.2008.01.002.

Figure 1.  Graphs of $ F(\omega ) $ in black, $ G_{1}(\omega ) $ in red and $ G_{2}(\omega ) $ in blue
Figure 2.  Stability switching curves
Figure 3.  Dynamics at point $ a $ for various values of $ \tau _{1} $ and $ \tau _{2} $
Figure 4.  Dynamics at point $ b $ for various values of $ \tau _{1} $ and $ \tau _{2} $
Figure 5.  Dynamics at point $ c $ for various values of $ \tau _{2} $ and $ \tau _{3} $
Figure 6.  Graphs of $ F(\omega ) $ in black, $ G_{1}(\omega ) $ in red and $ G_{2}(\omega ) $ in blue
Figure 7.  Stability switching curves
Figure 8.  Nonlinear adjustment coefficient functions
[1]

Rabah Amir, Igor V. Evstigneev. A new perspective on the classical Cournot duopoly. Journal of Dynamics and Games, 2017, 4 (4) : 361-367. doi: 10.3934/jdg.2017019

[2]

Iraklis Kollias, Elias Camouzis, John Leventides. Global analysis of solutions on the Cournot-Theocharis duopoly with variable marginal costs. Journal of Dynamics and Games, 2017, 4 (1) : 25-39. doi: 10.3934/jdg.2017002

[3]

Rafael Ortega. Stability and index of periodic solutions of a nonlinear telegraph equation. Communications on Pure and Applied Analysis, 2005, 4 (4) : 823-837. doi: 10.3934/cpaa.2005.4.823

[4]

Ismail Abdulrashid, Abdallah A. M. Alsammani, Xiaoying Han. Stability analysis of a chemotherapy model with delays. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 989-1005. doi: 10.3934/dcdsb.2019002

[5]

Litan Yan, Wenyi Pei, Zhenzhong Zhang. Exponential stability of SDEs driven by fBm with Markovian switching. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6467-6483. doi: 10.3934/dcds.2019280

[6]

Bin Fang, Xue-Zhi Li, Maia Martcheva, Li-Ming Cai. Global stability for a heroin model with two distributed delays. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 715-733. doi: 10.3934/dcdsb.2014.19.715

[7]

Yukihiko Nakata, Yoichi Enatsu, Yoshiaki Muroya. On the global stability of an SIRS epidemic model with distributed delays. Conference Publications, 2011, 2011 (Special) : 1119-1128. doi: 10.3934/proc.2011.2011.1119

[8]

Marek Bodnar, Monika Joanna Piotrowska, Urszula Foryś, Ewa Nizińska. Model of tumour angiogenesis -- analysis of stability with respect to delays. Mathematical Biosciences & Engineering, 2013, 10 (1) : 19-35. doi: 10.3934/mbe.2013.10.19

[9]

Desheng Li, P.E. Kloeden. Robustness of asymptotic stability to small time delays. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 1007-1034. doi: 10.3934/dcds.2005.13.1007

[10]

Ying Sue Huang, Chai Wah Wu. Stability of cellular neural network with small delays. Conference Publications, 2005, 2005 (Special) : 420-426. doi: 10.3934/proc.2005.2005.420

[11]

Miljana Jovanović, Vuk Vujović. Stability of stochastic heroin model with two distributed delays. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2407-2432. doi: 10.3934/dcdsb.2020016

[12]

Ábel Garab, Veronika Kovács, Tibor Krisztin. Global stability of a price model with multiple delays. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6855-6871. doi: 10.3934/dcds.2016098

[13]

Hermann Brunner, Chunhua Ou. On the asymptotic stability of Volterra functional equations with vanishing delays. Communications on Pure and Applied Analysis, 2015, 14 (2) : 397-406. doi: 10.3934/cpaa.2015.14.397

[14]

Zhen Jin, Zhien Ma. The stability of an SIR epidemic model with time delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 101-109. doi: 10.3934/mbe.2006.3.101

[15]

Gerhard Keller. Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors. Discrete and Continuous Dynamical Systems - S, 2017, 10 (2) : 313-334. doi: 10.3934/dcdss.2017015

[16]

Cruz Vargas-De-León, Alberto d'Onofrio. Global stability of infectious disease models with contact rate as a function of prevalence index. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1019-1033. doi: 10.3934/mbe.2017053

[17]

Paul Georgescu, Hong Zhang, Daniel Maxin. The global stability of coexisting equilibria for three models of mutualism. Mathematical Biosciences & Engineering, 2016, 13 (1) : 101-118. doi: 10.3934/mbe.2016.13.101

[18]

Mustapha Ait Rami, Vahid S. Bokharaie, Oliver Mason, Fabian R. Wirth. Stability criteria for SIS epidemiological models under switching policies. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2865-2887. doi: 10.3934/dcdsb.2014.19.2865

[19]

Roberto Avanzi, Nicolas Thériault. A filtering method for the hyperelliptic curve index calculus and its analysis. Advances in Mathematics of Communications, 2010, 4 (2) : 189-213. doi: 10.3934/amc.2010.4.189

[20]

Serge Nicaise, Julie Valein, Emilia Fridman. Stability of the heat and of the wave equations with boundary time-varying delays. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 559-581. doi: 10.3934/dcdss.2009.2.559

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (192)
  • HTML views (290)
  • Cited by (0)

Other articles
by authors

[Back to Top]