Complex Dynamics of Bertrand Duopoly Games with Bounded Rationality
A dynamic of Bertrand duopoly game is analyzed, where players use different production methods and choose their prices with bounded rationality. The equilibriums of the corresponding discrete dynamical systems are investigated. The stability conditions of Nash equilibrium under a local adjustment process are studied. The stability conditions of Nash equilibrium under a local adjustment process are studied. The stability of Nash equilibrium, as some parameters of the model are varied, gives rise to complex dynamics such as cycles of higher order and chaos. On this basis, we discover that an increase of adjustment speed of bounded rational player can make Bertrand market sink into the chaotic state. Finally, the complex dynamics, bifurcations and chaos are displayed by numerical simulation.
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Savants, pp.499-508, 1883.
[2] H.Gravelle,R.Rees. Microeconomics. 2nd edition, Harlow:
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Basel: Birkhauser,1999.
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published.
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costs. Economics Letters, vol. 57, no. 1, pp. 87-96,1997.
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vol. 17, no. 4, pp.513-525, 1999.
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an experimental study. International Journal of Industrial Organization,
vol. 18, no. 7, pp. 7-22, 2002..
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[10] C.F.Lo, D.Kiang. Quantum Bertrand duopoly with differentiated
products. Physics Letters A, vol. 321, no. 2, pp. 94-98,2004.
[11] Tao huang.Bertrand competition with incomplete share for lower price.
Economics Letters, vol.83, no. 2, pp. 239-244,2004.
[12] Xiaohang Yue, Samar K. Mukhopadhyay, Xiaowei Zhu. A Bertrand
model of pricing of complementary goods under information asymmetry.
Journal of Business Research, vol.59, no.10, pp. 1182-1192,2006.
[13] Jason J. Lepore. Cournot and Bertrand–Edgeworth competition when
rivals' costs are unknown. Economics Letters, vol. 103, no. 3, pp.
237-240,2007..
[14] Ola Andersson. On the role of patience in collusive Bertrand duopolies.
Economics Letters, vol. 100, no. 1, pp. 60-63,2007.
[15] Fernanda A. Ferreira, Flávio Ferreiraa. Maximum revenue tariff under
Bertrand duopoly with unknown costs(J). Communications in Nonlinear
Science and Numerical Simulation, to be published.
[16] H.S. Bierman, L. Fernandez. Game Theory with Economic Applications,
2nd Edition, Reading, Mass: Addison-Wesley, 1998.
[17] Dixit,A. Comparative statics for oligopoly. International Economic
Review, vol.100, no.1, pp.107-122,1986.
[18] Henon,M. A two dimensional mapping with a strange attractor.
Communications in Mathematical Physics, no. 50, pp.69-77,1976..
[19] Hubertus F. von Bremen, Firdaus E. Udwadia, Wlodek Proskurowski. An
efficient QR based method for the computation of Lyapunov exponents.
Physica D, vol. 101, no. 1, pp.1-16,1997.
[20] James L. Kaplan, James A. Yorke. A regime observed in a fluid flow
model of Lorenz. Communications in Mathematical Physics , vol. 67, no.
1, pp. 93-108,1979.</p>
<p>[1] Bertrand,J. Theories Mathematique de la Richesse Social.Journal des
Savants, pp.499-508, 1883.
[2] H.Gravelle,R.Rees. Microeconomics. 2nd edition, Harlow:
Longman,1992.
[3] Agiza,H.N., Elsadany,A.A. Chaotic dynamics in nonlinear duopoly game
with heterogeneous players. Applied Mathematics and Computation, vol.
149, no. 3, pp. 843-860,2004.
[4] Bischi G I, Naimzada A. Advances in Dynamic Games and Application.
Basel: Birkhauser,1999.
[5] Jixiang Zhang, Qingli Da, Yanhua Wang. The dynamics of Bertrand
model with bounded rationality. Chaos, Solitons and Fractals, to be
published.
[6] Robert Marquez. A note on Bertrand competition with asymmetric fixed
costs. Economics Letters, vol. 57, no. 1, pp. 87-96,1997.
[7] Walter Elberfeld, Elmar Wolfstetter. A dynamic model of Bertrand
competition with entry. International Journal of Industrial Organization,
vol. 17, no. 4, pp.513-525, 1999.
[8] Dufwenberg.M, Gneezy.U. Price competition and market concentration:
an experimental study. International Journal of Industrial Organization,
vol. 18, no. 7, pp. 7-22, 2002..
[9] Alós–Ferrer, Ana.B.Ania and Klaus Reiner Schenk Hoppé. An
Evolutionary Model of Bertrand Oligopoly(J). Games and Economic
Behavior, vol. 33, no. 1, pp. 1-19,2000.
[10] C.F.Lo, D.Kiang. Quantum Bertrand duopoly with differentiated
products. Physics Letters A, vol. 321, no. 2, pp. 94-98,2004.
[11] Tao huang.Bertrand competition with incomplete share for lower price.
Economics Letters, vol.83, no. 2, pp. 239-244,2004.
[12] Xiaohang Yue, Samar K. Mukhopadhyay, Xiaowei Zhu. A Bertrand
model of pricing of complementary goods under information asymmetry.
Journal of Business Research, vol.59, no.10, pp. 1182-1192,2006.
[13] Jason J. Lepore. Cournot and Bertrand–Edgeworth competition when
rivals' costs are unknown. Economics Letters, vol. 103, no. 3, pp.
237-240,2007..
[14] Ola Andersson. On the role of patience in collusive Bertrand duopolies.
Economics Letters, vol. 100, no. 1, pp. 60-63,2007.
[15] Fernanda A. Ferreira, Flávio Ferreiraa. Maximum revenue tariff under
Bertrand duopoly with unknown costs(J). Communications in Nonlinear
Science and Numerical Simulation, to be published.
[16] H.S. Bierman, L. Fernandez. Game Theory with Economic Applications,
2nd Edition, Reading, Mass: Addison-Wesley, 1998.
[17] Dixit,A. Comparative statics for oligopoly. International Economic
Review, vol.100, no.1, pp.107-122,1986.
[18] Henon,M. A two dimensional mapping with a strange attractor.
Communications in Mathematical Physics, no. 50, pp.69-77,1976..
[19] Hubertus F. von Bremen, Firdaus E. Udwadia, Wlodek Proskurowski. An
efficient QR based method for the computation of Lyapunov exponents.
Physica D, vol. 101, no. 1, pp.1-16,1997.
[20] James L. Kaplan, James A. Yorke. A regime observed in a fluid flow
model of Lorenz. Communications in Mathematical Physics , vol. 67, no.
1, pp. 93-108,1979.</p>
@article{"International Journal of Business, Human and Social Sciences:63995", author = "Jixiang Zhang and Guocheng Wang", title = "Complex Dynamics of Bertrand Duopoly Games with Bounded Rationality", abstract = "A dynamic of Bertrand duopoly game is analyzed, where players use different production methods and choose their prices with bounded rationality. The equilibriums of the corresponding discrete dynamical systems are investigated. The stability conditions of Nash equilibrium under a local adjustment process are studied. The stability conditions of Nash equilibrium under a local adjustment process are studied. The stability of Nash equilibrium, as some parameters of the model are varied, gives rise to complex dynamics such as cycles of higher order and chaos. On this basis, we discover that an increase of adjustment speed of bounded rational player can make Bertrand market sink into the chaotic state. Finally, the complex dynamics, bifurcations and chaos are displayed by numerical simulation.
", keywords = "Bertrand duopoly model, Discrete dynamical system,
Heterogeneous expectations, Nash equilibrium.", volume = "7", number = "7", pages = "2181-5", }
