Dynamics of a Bertrand Duopoly Game of the Greek Oil Market and Application of the d-Backtest Method
Abstract: This work presents the complexity that emerges in a Bertrand duopoly between two companies in the Greek oil market, one of which is semi-public and the other is private. The game uses linear demand functions for differentiated products from the existing literature and asymmetric cost functions that arose after approaches using the published financial reports of the two oil companies (Hellenic Petroleum and Motor Oil). The game is based on the assumption of homogeneous players who are characterized by bounded rationality and follow an adjustment mechanism. The players’ decisions for each time period are expressed by two difference equations. A dynamical analysis of the game’s discrete dynamical system is made by finding the equilibrium positions and studying their stability. Numerical simulations include bifurcation diagrams and strange attractors. Lyapunov numbers’ graphs and sensitivity analysis in initial conditions prove the algebraic results and reveal the complexity and chaotic behavior of the system focusing on the two parameters k1 and k2 (speed of adjustment for each player). The d-Backtest method is applied through which an attempt is made to control the chaos that appears outside the stability space in order to return to the locally asymptotically stable Nash equilibrium for the system.
Cite this paper: Sarafopoulos, G. , Drimpetas, E. , Papadopoulos, K. and Vezeris, D. (2021) Dynamics of a Bertrand Duopoly Game of the Greek Oil Market and Application of the d-Backtest Method. Applied Mathematics, 12, 1097-1117. doi: 10.4236/am.2021.1211070.
References

[1]   Agiza, H.N. (1999) On the Analysis of Stability, Bifurcation, Chaos and Chaos Control of Kopel Map. Chaos, Solitons & Fractals, 10, 1909-1916.
https://doi.org/10.1016/S0960-0779(98)00210-0

[2]   Agiza, H.N., Hegazi, A.S. and Elsadany, A.A. (2002) Complex Dynamics and Synchronization of Duopoly Game with Bounded Rationality. Mathematics and Computers in Simulation, 58, 133-146.
https://doi.org/10.1016/S0378-4754(01)00347-0

[3]   Agliar, A., Gardini, L. and Puu, T. (2005) Some Global Bifurcations Related to the Appearance of Closed Invariant Curves. Mathematics and Computers in Simulation, 68, 201-219.
https://doi.org/10.1016/j.matcom.2004.12.003

[4]   Agliari, A., Gardini, L. and Puu, T. (2006) Global Bifurcations in Duopoly When the Cournot Point Is Destabilized via a Subcritical Neimark Bifurcation. International Game Theory Review, 8, 1-20.
https://doi.org/10.1142/S0219198906000758

[5]   Bishi, G.I. and Kopel, M. (2001) Equilibrium Selection in a Nonlinear Duopoly Game with Adaptive Expectations. Journal of Economic Behavior & Organization, 46, 73-100.
https://doi.org/10.1016/S0167-2681(01)00188-3

[6]   Kopel, M. (1996) Simple and Complex Adjustment Dynamics in Cournot Duopoly Models. Chaos, Solitons & Fractals, 7, 2031-2048.
https://doi.org/10.1016/S0960-0779(96)00070-7

[7]   Puu, T. (1998) The Chaotic Duopolists Revisited. Journal of Economic Beahavior & Organization, 33, 385-394.
https://doi.org/10.1016/S0167-2681(97)00064-4

[8]   Puu, T. (2005) Complex Oligopoly Dynamics. In: Lines, M., Ed., Nonlinear Dynamical Systems in Economics, Springer, New York, 165-186.
https://doi.org/10.1007/3-211-38043-4_6

[9]   Sarafopoulos, G. (2015) On the Dynamics of a Duopoly Game with Differentiated Goods. Procedia Economics and Finance, 19, 146-153.
https://doi.org/10.1016/S2212-5671(15)00016-7

[10]   Sarafopoulos, G. (2015) Complexity in a Duopoly Game with Homogeneous Players, Convex, Log-Linear Demand and Quadratic Cost Functions. Procedia Economics and Finance, 33, 358-366.
https://doi.org/10.1016/S2212-5671(15)01720-7

[11]   Zhang, J., Da, Q. and Wang, Y. (2009) The Dynamics of Bertrand Model with Bounded Rationality. Chaos, Solitons and Fractals, 39, 2048-2055.
https://doi.org/10.1016/j.chaos.2007.06.056

[12]   Agiza, H.N. and Elsadany, A.A. (2003) Nonlinear Dynamics in the Cournot Duopoly Game with Heterogeneous Players. Physica A, 320, 512-524.
https://doi.org/10.1016/S0378-4371(02)01648-5

[13]   Agiza, H.N. and Elsadany, A.A. (2004) Chaotic Dynamics in Nonlinear Duopoly Game with Heterogenous Players. Applied Mathematics and Computation, 149, 843-860.
https://doi.org/10.1016/S0096-3003(03)00190-5

[14]   Haan, W.J.D. (2001) The Importance of the Number of Different Agents in a Heterogeneous Asset—Pricing Model. Journal of Economic Dynamic and Control, 25, 721-746.
https://doi.org/10.1016/S0165-1889(00)00038-5

[15]   Hommes, C.H. (2006) Heterogeneous Agent Models in Economics and Finance. In: Tesfatsion, L. and Judd, K.L., Eds., Handbook of Computational Economics, Agent-Based Computational Economics, Vol. 2, Elsevier Science B.V., Amsterdam, 1109-1186.
https://doi.org/10.1016/S1574-0021(05)02023-X

[16]   Fanti, L. and Gori, L. (2012) The Dynamics of a Differentiated Duopoly with Quantity Competition. Economic Modelling, 29, 421-427.
https://doi.org/10.1016/j.econmod.2011.11.010

[17]   Gao, Y. (2009) Complex Dynamics in Two Dimensional Noninvertible Map. Chaos Solitons & Fractals, 39, 1798-1810.
https://doi.org/10.1016/j.chaos.2007.06.051

[18]   Sarafopoulos, G. and Papadopoulos, K. (2017) On a Cournot Duopoly Game with Differentiated Goods, Heterogeneous Expectations and a Cost Function Including Emission Costs. Scientific Bulletin—Economic Science, 16, 11-22.
https://econpapers.repec.org/article/ptsjournl/y_3a2017_3ai_3a1_3ap_3a11-22.htm

[19]   Sarafopoulos, G. and Papadopoulos, K. (2019) Complexity in a Bertrand Duopoly Game with Heterogeneous Players and Differentiated Goods. In: Sykianakis, N., Polychronidou, P. and Karasavvoglou, A., Eds., Economic and Financial Challenges for Eastern Europe, Springer, Berlin, 15-26.
https://doi.org/10.1007/978-3-030-12169-3_2

[20]   Sarafopoulos, G. and Papadopoulos, K. (2020) On a Bertrand Dynamic Game with Differentiated Goods, Heterogeneous Expectations and Asymmetric Cost Functions. In: Janowicz-Lomott, M., Łyskawa, K. and Polychronidou, P, Eds., Economic and Financial Challenges for Balkan and Eastern European Countries, Springer, Berlin, 223-241.
https://doi.org/10.1007/978-3-030-39927-6_14

[21]   Tramontana, F. (2010) Heterogeneous Duopoly with Isoelastic Demand Function. Economic Modelling, 27, 350-357.
https://doi.org/10.1016/j.econmod.2009.09.014

[22]   Zhang, J., Da, Q. and Wang, Y. (2007) Analysis of Nonlinear Duopoly Game with Heterogeneous Players. Economic Modelling, 24, 138-148.
https://doi.org/10.1016/j.econmod.2006.06.007

[23]   Wu, W., Chen, Z. and Ip, W.H. (2010) Complex Nonlinear Dynamics and Controlling Chaos in a Cournot Duopoly Economic Model. Nonlinear Analysis: Real World Applications, 11, 4363-4377.
https://doi.org/10.1016/j.nonrwa.2010.05.022

[24]   Baumol, W.J. and Quandt, R.E. (1964) Rules of Thumb and Optimally Imperfect Decisions. American Economic Review, 54, 23-46.
https://www.jstor.org/stable/1810896

[25]   Singh, N. and Vives, X. (1984) Price and Quantity Competition in a Differentiated Duopoly. The RAND Journal of Economics, 15, 546-554.
https://doi.org/10.2307/2555525

[26]   Puu, T. (1991) Chaos in Duopoly Pricing. Chaos, Solitons & Fractals, 1, 573-581.
https://doi.org/10.1016/0960-0779(91)90045-B

[27]   Puu, T. (1995) The Chaotic Monopolist. Chaos, Solitons & Fractals, 5, 35-44.
https://doi.org/10.1016/0960-0779(94)00206-6

[28]   Westerhoff, F. (2006) Nonlinear Expectation Formation, Endogenous Business Cycles and Stylized Facts. Studies in Nonlinear Dynamics and Econometrics, 10, Article No. 4.
https://doi.org/10.2202/1558-3708.1324

[29]   Naimzada, A.K. and Ricchiuti, G. (2008) Complex Dynamics in a Monopoly with a Rule of Thumb. Applied Mathematics and Computation, 203, 921-925.
https://doi.org/10.1016/j.amc.2008.04.020

[30]   Asksar, S.S. (2013) On Complex Dynamics of Monopoly Market. Economic Modelling, 31, 586-589.
https://doi.org/10.1016/j.econmod.2012.12.025

[31]   Askar, S.S. (2014) Complex Dynamic Properties of Cournot Duopoly Games with Convex and Log-Concave Demand Function. Operations Research Letters, 42, 85-90.
https://doi.org/10.1016/j.orl.2013.12.006

[32]   Elsadany, A. and Awad, A.M. (2016) Nonlinear Dynamics of Cournot Duopoly Game with Social Welfare. Electronic Journal of Mathematician Analysis and Applications, 4, 173-191.
http://math-frac.org/Journals/EJMAA/Vol4(2)_July_2016/Vol4(2)_Papers/16_EJMAA_Vol4(2)_July_2016_pp_173-191.pdf

[33]   Naimzada, A.K. and Sbragia, L. (2006) Oilgopoly Games with Nonlinear Demand and Cost Functions: Two Boundedly Rational Adjustment Processes. Chaos, Solitons & Fractals, 29, 707-722.
https://doi.org/10.1016/j.chaos.2005.08.103

[34]   Gandolfo, G. (1997) Economic Dynamics. Springer, Berlin.
https://www.springer.com/gp/book/9783642038624

[35]   Elaydi, S. (2005) An Introduction to Difference Equations. 3rd Edition, Springer-Verlag, New York.

[36]   Kulenovic, M. and Merino, O. (2002) Discrete Dynamical Systems and Difference Equations with Mathematica. Chapman & Hall/CRC, London.
https://doi.org/10.1201/9781420035353

[37]   Vezeris, D., Schinas, C. and Papaschinopoulos, G. (2018) Profitability Edge by Dynamic Back Testing Optimal Period Selection Technical Parameters Optimization in Trading Systems with Forecasting. Computational Economics, 51, 761-807.
https://doi.org/10.1007/s10614-016-9640-x

[38]   Vezeris, D., Schinas, C., Kyrgos, T., Bizergianidou, V. and Karkanis, I. (2019) Optimizations of Backtesting Techniques in Automated High Frequency Trading Systems Using the d-Backtest PS Method. Computational Economics, 56, 975-1054.
https://doi.org/10.1007/s10614-019-09956-1

[39]   Vezeris, D., Kyrgos, T., Karkanis, I. and Bizergianidou, V. (2020) Automated Trading Systems’ Evaluation Using d-Backtest PS Method and WM Ranking in Financial Markets. Investment Management and Financial Innovations, 17, 198-215.
https://doi.org/10.21511/imfi.17(2).2020.16

[40]   Sarafopoulos, G., Drimpetas, E., Papadopoulos, K. and Vezeris, D. (2021) Chaotic Behavior in Duopoly Market and Application of the d-Backtest Method. 14th Chaotic Modeling and Simulation International Conference CHAOS 2021, Athens, 8-11 June 2021.

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