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## 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  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021069
##### References:
 [1] R. Bellman and K. L. Cooke, Difference-Differential Equations, Academic Press, New York, 1963.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [5] J. W. Friedman, Game Theory with Applications to Economics, Oxford University Press, New York, Oxford, 1986.   Google Scholar [6] L. Gori, L. 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.  Google Scholar [7] K. Gu, S. 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.  Google Scholar [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.  Google Scholar [9] F. Hahn, The stability of the Cournot oligopoly solution, Review of Economic Studies, 29 (1962), 329-331.   Google Scholar [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.  Google Scholar [11] X. Lin and H. Wang, Stability analysis of delay differential equations with two discrete delays, Canadian Applied Mathematics Quarterly, 20 (2012), 519-533.   Google Scholar [12] A. Matsumoto and F. Szidarovszky, Dynamic Oligopolies with Time Delays, Springer Nature, Singapore, 2018. doi: 10.1007/978-981-13-1786-6.  Google Scholar [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.  Google Scholar [14] A. Matsumoto, F. Szidarovszky and H. Yosida, Dynamics in linear Cournot duopolies with two time delays, Computational Economics, 38 (2011), 311-327.   Google Scholar [15] M. McManus and R. Quandt, Comments on the stability of the Cournot oligopoly model, Review of Economic Studies, 28 (1964), 136-139.   Google Scholar [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.  Google Scholar [17] K. Okuguchi, Expectations and Stability in Oligopoly Models, Springer, Berlin, 1976.  Google Scholar [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.  Google Scholar [19] J. B. Rosen, Existence and uniqueness of equilibrium points for concave $n$-person games, Econometrica, 33 (1965), 520-534.  doi: 10.2307/1911749.  Google Scholar [20] Y. Song, M. 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.  Google Scholar [21] F. Szidarovszky, On the oligopoly game, DM-70-1, Karl Marx University of Economics, Budapest, Hungary, 1970.  Google Scholar [22] R. Theocharis, On the stability of the Cournot solution on the oligopoly problem, Review of Economic Studies, 27 (1960), 133-134.   Google Scholar [23] J. Zhang, Z. Jin, J. 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.  Google Scholar

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##### References:
 [1] R. Bellman and K. L. Cooke, Difference-Differential Equations, Academic Press, New York, 1963.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [5] J. W. Friedman, Game Theory with Applications to Economics, Oxford University Press, New York, Oxford, 1986.   Google Scholar [6] L. Gori, L. 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.  Google Scholar [7] K. Gu, S. 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.  Google Scholar [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.  Google Scholar [9] F. Hahn, The stability of the Cournot oligopoly solution, Review of Economic Studies, 29 (1962), 329-331.   Google Scholar [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.  Google Scholar [11] X. Lin and H. Wang, Stability analysis of delay differential equations with two discrete delays, Canadian Applied Mathematics Quarterly, 20 (2012), 519-533.   Google Scholar [12] A. Matsumoto and F. Szidarovszky, Dynamic Oligopolies with Time Delays, Springer Nature, Singapore, 2018. doi: 10.1007/978-981-13-1786-6.  Google Scholar [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.  Google Scholar [14] A. Matsumoto, F. Szidarovszky and H. Yosida, Dynamics in linear Cournot duopolies with two time delays, Computational Economics, 38 (2011), 311-327.   Google Scholar [15] M. McManus and R. Quandt, Comments on the stability of the Cournot oligopoly model, Review of Economic Studies, 28 (1964), 136-139.   Google Scholar [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.  Google Scholar [17] K. Okuguchi, Expectations and Stability in Oligopoly Models, Springer, Berlin, 1976.  Google Scholar [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.  Google Scholar [19] J. B. Rosen, Existence and uniqueness of equilibrium points for concave $n$-person games, Econometrica, 33 (1965), 520-534.  doi: 10.2307/1911749.  Google Scholar [20] Y. Song, M. 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.  Google Scholar [21] F. Szidarovszky, On the oligopoly game, DM-70-1, Karl Marx University of Economics, Budapest, Hungary, 1970.  Google Scholar [22] R. Theocharis, On the stability of the Cournot solution on the oligopoly problem, Review of Economic Studies, 27 (1960), 133-134.   Google Scholar [23] J. Zhang, Z. Jin, J. 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.  Google Scholar
Graphs of $F(\omega )$ in black, $G_{1}(\omega )$ in red and $G_{2}(\omega )$ in blue
Stability switching curves
Dynamics at point $a$ for various values of $\tau _{1}$ and $\tau _{2}$
Dynamics at point $b$ for various values of $\tau _{1}$ and $\tau _{2}$
Dynamics at point $c$ for various values of $\tau _{2}$ and $\tau _{3}$
Graphs of $F(\omega )$ in black, $G_{1}(\omega )$ in red and $G_{2}(\omega )$ in blue
Stability switching curves