A Bertrand duopoly game with long-memory effects. (English) Zbl 1445.91029
Summary: Reconsidering the Bertrand duopoly game based on the concept of long short-term memory, we construct a fractional-order Bertrand duopoly game by extending the integer-order game to its corresponding fractional-order form. We build such a Bertrand duopoly game, in which both players can make their decisions with long-memory effects. Then, we investigate its Nash equilibria, local stability, and numerical solutions. Using the bifurcation diagram, the phase portrait, time series, and the 0-1 test for chaos, we numerically validate these results and illustrate its complex phenomena, such as bifurcation and chaos.
MSC:
| 91B54 | Special types of economic markets (including Cournot, Bertrand) |
| 91B55 | Economic dynamics |
| 91A25 | Dynamic games |
References:
| [1] | Ueda, M., Effect of information asymmetry in Cournot duopoly game with bounded rationality, Applied Mathematics and Computation, 362, 8 (2019) · Zbl 1433.91047 · doi:10.1016/j.amc.2019.06.049 |
| [2] | Tu, H.; Wang, X., Research on a dynamic master-slave Cournot triopoly game model with bounded rational rule and its control, Mathematical Problems in Engineering, 2016 (2016) · Zbl 1400.91095 · doi:10.1155/2016/2353909 |
| [3] | Hsu, J.; Liu, L.; Henry Wang, X.; Zeng, C., Ad valorem versus per-unit royalty licensing in a Cournot duopoly model, The Manchester School, 87, 6, 890-901 (2019) · doi:10.1111/manc.12280 |
| [4] | Al-khedhairi, A., Differentiated Cournot duopoly game with fractional-order and its discretization, Engineering Computations, 36, 3, 781-806 (2019) · doi:10.1108/ec-07-2018-0333 |
| [5] | Xin, B.; Wang, Y., Stability and Hopf bifurcation of a stochastic Cournot duopoly game in a blockchain cloud services market driven by brownian motion, IEEE Access, 8, 41432-41438 (2020) · doi:10.1109/access.2020.2976501 |
| [6] | Zhou, J.; Zhou, W.; Chu, T.; Chang, Y.-X.; Huang, M.-J., Bifurcation, intermittent chaos and multi-stability in a two-stage Cournot game with R&D spillover and product differentiation, Applied Mathematics and Computation, 341, 358-378 (2019) · Zbl 1429.91203 · doi:10.1016/j.amc.2018.09.004 |
| [7] | Zhang, Y.-h.; Zhou, W.; Chu, T.; Chu, Y.-D.; Yu, J.-N., Complex dynamics analysis for a two-stage Cournot duopoly game of semi-collusion in production, Nonlinear Dynamics, 91, 2, 819-835 (2018) · Zbl 1390.91238 · doi:10.1007/s11071-017-3912-4 |
| [8] | Gori, L.; Sodini, M., Price competition in a nonlinear differentiated duopoly, Chaos, Solitons & Fractals, 104, 557-567 (2017) · Zbl 1380.91099 · doi:10.1016/j.chaos.2017.09.009 |
| [9] | Askar, S. S.; Al-khedhairi, A., Dynamic investigations in a duopoly game with price competition based on relative profit and profit maximization, Journal of Computational and Applied Mathematics, 367 (2020) · Zbl 1427.91157 · doi:10.1016/j.cam.2019.112464 |
| [10] | Xin, B.; Peng, W.; Guerrini, L., A continuous time Bertrand duopoly game with fractional delay and conformable derivative: modelling, discretization process, Hopf bifurcation and chaos, Frontiers in Physics, 7, 84 (2019) · doi:10.3389/fphy.2019.00084 |
| [11] | Ma, J.; Lou, W., Complex characteristics of multichannel household appliance supply chain with the price competition, Complexity, 2017 (2017) · Zbl 1367.93055 · doi:10.1155/2017/4327069 |
| [12] | Fanti, L.; Gori, L.; Mammana, C.; Michetti, E., The dynamics of a Bertrand duopoly with differentiated products: synchronization, intermittency and global dynamics, Chaos, Solitons & Fractals, 52, 73-86 (2013) · Zbl 1323.37052 · doi:10.1016/j.chaos.2013.04.002 |
| [13] | Tu, H.; Zhan, X.; Mao, X., Complex dynamics and chaos control on a kind of Bertrand duopoly game model considering R&D activities, Discrete Dynamics in Nature and Society, 2017 (2017) · Zbl 1412.91080 · doi:10.1155/2017/7384150 |
| [14] | Ahmed, E.; Elsadany, A. A.; Puu, T., On Bertrand duopoly game with differentiated goods, Applied Mathematics and Computation, 251, 169-179 (2015) · Zbl 1328.91202 · doi:10.1016/j.amc.2014.11.051 |
| [15] | Yang, X.; Peng, Y.; Xiao, Y.; Wu, X., Nonlinear dynamics of a duopoly Stackelberg game with marginal costs, Chaos, Solitons & Fractals, 123, 185-191 (2019) · Zbl 1448.91071 · doi:10.1016/j.chaos.2019.04.007 |
| [16] | Peng, Y.; Lu, Q., Complex dynamics analysis for a duopoly Stackelberg game model with bounded rationality, Applied Mathematics and Computation, 271, 259-268 (2015) · Zbl 1410.91137 · doi:10.1016/j.amc.2015.08.138 |
| [17] | Shi, L.; Le, Y.; Sheng, Z., Analysis of price Stackelberg duopoly game with bounded rationality, Discrete Dynamics in Nature and Society, 2014 (2014) · Zbl 1422.91478 · doi:10.1155/2014/428568 |
| [18] | Dong, Y.; Zhang, Y.; Pan, J.; Chen, T., Evolutionary game model of stock price synchronicity from investor behavior, Discrete Dynamics in Nature and Society, 2020 (2020) · Zbl 1459.91227 · doi:10.1155/2020/7957282 |
| [19] | Ma, J.; Sun, L.; Hou, S.; Zhan, X., Complexity study on the Cournot-Bertrand mixed duopoly game model with market share preference, Chaos: An Interdisciplinary Journal of Nonlinear Science, 28, 2 (2018) · Zbl 1391.91121 · doi:10.1063/1.5001353 |
| [20] | Wu, C.-H., Licensing to a competitor and strategic royalty choice in a dynamic duopoly, European Journal of Operational Research, 279, 3, 840-853 (2019) · Zbl 1431.91224 · doi:10.1016/j.ejor.2019.06.035 |
| [21] | Xin, B.; Peng, W.; Kwon, Y., A Fractional-Order Difference Cournot Duopoly Game with Long Memory (2019), Ithaca, NY, USA: Cornell University Press, Ithaca, NY, USA |
| [22] | Bischi, G. I.; Naimzada, A., Global analysis of a dynamic duopoly game with bounded rationality, Advances in Dynamic Games and Applications, 361-385 (2000), Birkhuser Boston: Springer, Birkhuser Boston · Zbl 0957.91027 · doi:10.1007/978-1-4612-1336-9_20 |
| [23] | Cermak, J.; Gyori, I.; Nechvatal, L., On explicit stability conditions for a linear fractional difference system, Fractional Calculus and Applied Analysis, 18, 3, 651-672 (2015) · Zbl 1320.39004 · doi:10.1515/fca-2015-0040 |
| [24] | Chen, F.; Luo, X.; Zhou, Y., Existence results for nonlinear fractional difference equation, Advances in Difference Equations, 2011, 1, 1-12 (2011) · Zbl 1207.39012 · doi:10.1155/2011/713201 |
| [25] | Wu, G.-C.; Baleanu, D., Discrete fractional logistic map and its chaos, Nonlinear Dynamics, 75, 1-2, 283-287 (2014) · Zbl 1281.34121 · doi:10.1007/s11071-013-1065-7 |
| [26] | Gottwald, G. A.; Melbourne, I., On the validity of the 0-1 test for chaos, Nonlinearity, 22, 6, 1367-1382 (2009) · Zbl 1171.65091 · doi:10.1088/0951-7715/22/6/006 |
| [27] | Falconer, I.; Gottwald, G. A.; Melbourne, I.; Wormnes, K., Application of the 0-1 test for chaos to experimental data, SIAM Journal on Applied Dynamical Systems, 6, 2, 395-402 (2007) · Zbl 1210.37059 · doi:10.1137/060672571 |
| [28] | Yuan, L.; Zheng, S.; Alam, Z., Dynamics analysis and cryptographic application of fractional logistic map, Nonlinear Dynamics, 96, 1, 615-636 (2019) · Zbl 1437.94079 · doi:10.1007/s11071-019-04810-3 |
| [29] | Chen, T.; Xiao, B.; Liu, H., Credit risk contagion in an evolving network model integrating spillover effects and behavioral interventions, Complexity, 2018 (2018) · Zbl 1398.91637 · doi:10.1155/2018/1843792 |
| [30] | Chen, T.; Wang, J.; Liu, H.; He, Y., Contagion model on counterparty credit risk in the crt market by considering the heterogeneity of counterparties and preferential-random mixing attachment, Physica A: Statistical Mechanics and Its Applications, 520, 458-480 (2019) · doi:10.1016/j.physa.2019.01.018 |
| [31] | Chen, T.; Wang, Y.; Zeng, Q.; Luo, J., Network model of credit risk contagion in the interbank market by considering bank runs and the fire sale of external assets, Physica A: Statistical Mechanics and Its Applications, 542 (2020) · Zbl 1539.91137 · doi:10.1016/j.physa.2019.123006 |
| [32] | Xin, B.; Sun, M., A differential oligopoly game for optimal production planning and water savings, European Journal of Operational Research, 269, 1, 206-217 (2018) · Zbl 1431.91225 · doi:10.1016/j.ejor.2017.07.016 |
| [33] | Xin, B.; Peng, W.; Kwon, Y.; Liu, Y., Modeling, discretization, and hyperchaos detection of conformable derivative approach to a financial system with market confidence and ethics risk, Advances in Difference Equations, 2019, 1, 138 (2019) · Zbl 1459.37095 · doi:10.1186/s13662-019-2074-8 |
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