# American Institute of Mathematical Sciences

August  2021, 26(8): 4549-4565. doi: 10.3934/dcdsb.2020302

## The effect of caputo fractional difference operator on a novel game theory model

 1 Laboratory of Dynamical Systems and Control, University of Larbi Ben M'hidi, Oum El Bouaghi, Algeria 2 Department of Mathematics, Faculty of Science, University of Ha'il, Kingdom of Saudi Arabia 3 Department of Mathematics and Computer Science, University of Larbi Ben M'hidi, Oum El Bouaghi, Algeria 4 Mathematics Department, Faculty of Science, University of Ha'il, Kingdom of Saudi Arabia 5 Universita del Salento, Dipartimento Ingegneria Innovazione, 73100 Lecce, Italy 6 Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

* Corresponding author: phamvietthanh@tdtu.edu.vn

Received  May 2020 Revised  August 2020 Published  August 2021 Early access  October 2020

It is well-known that fractional-order discrete-time systems have a major advantage over their integer-order counterparts, because they can better describe the memory characteristics and the historical dependence of the underlying physical phenomenon. This paper presents a novel fractional-order triopoly game with bounded rationality, where three firms producing differentiated products compete over a common market. The proposed game theory model consists of three fractional-order difference equations and is characterized by eight equilibria, including the Nash fixed point. When suitable values for the fractional order are considered, the stability of the Nash equilibrium is lost via a Neimark-Sacker bifurcation or via a flip bifurcation. As a consequence, a number of chaotic attractors appear in the system dynamics, indicating that the behaviour of the economic model becomes unpredictable, independently of the actions of the considered firm. The presence of chaos is confirmed via both the computation of the maximum Lyapunov exponent and the 0-1 test. Finally, an entropy algorithm is used to measure the complexity of the proposed game theory model.

Citation: Amina-Aicha Khennaoui, A. Othman Almatroud, Adel Ouannas, M. Mossa Al-sawalha, Giuseppe Grassi, Viet-Thanh Pham. The effect of caputo fractional difference operator on a novel game theory model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4549-4565. doi: 10.3934/dcdsb.2020302
##### References:
 [1] T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602-1611.  doi: 10.1016/j.camwa.2011.03.036. [2] A. Al-khedhairi, Differentiated Cournot duopoly game with fractional-order and its discretization, Engineering Computations, 36 (2019), 26. [3] A. Al-Khedhairi, Dynamics of a Cournot duopoly game with a generalized bounded rationality, Complexity, 2020 (2020), 8903183. [4] G. A. Anastassiou, Principles of delta fractional calculus on time scales and inequalities, Math. Comput. Model., 52 (2010), 556-566.  doi: 10.1016/j.mcm.2010.03.055. [5] S. S. Askar and A. Al-Khedhairi, Analysis of a four-firm competition based on a generalized bounded rationality and different mechanisms, Complexity, 2019 (2019), 6352796. doi: 10.1155/2019/6352796. [6] F. M. Atici and P. W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ. Spec. Ed. I, 3 (2009), 1-12.  doi: 10.14232/ejqtde.2009.4.3. [7] H. M. Baskonus, T. Mekkaoui, Z. Hammouch and H. Bulut, Active control of a chaotic fractional order economic system, Entropy, 17 (2015), 5771-5783.  doi: 10.3390/e17064255. [8] J. Cermak, I. Gyori and L. Nechvatal, On explicit stability conditions for a linear fractional difference system, Fractional Calculus and Applied Analysis, 18 (2015), 651-672. doi: 10.1515/fca-2015-0040. [9] G. A. Gottwald and I. Melbourne, On the implementation of the 0-1 test for chaos, SIAM J. Appl. Dyn. Syst., 8 (2009), 129-145.  doi: 10.1137/080718851. [10] N. Laskin, Fractional market dynamics, Physica A: Statist. Mech. Appl., 287 (2000), 482-492.  doi: 10.1016/S0378-4371(00)00387-3. [11] Y. Li, C. Sun, H. Ling, A. Lu and Y. Liu, Oligopolies price game in fractional order system, Chaos, Solitons & Fractals, 132 (2020), 109583. doi: 10.1016/j.chaos.2019.109583. [12] S. M. Pincus, Approximate entropy as a measure of system complexity, Proc. Natl. Acad. Sci. USA., 88 (1991), 2297–2301. doi: 10.1073/pnas.88.6.2297. [13] S. M. Pincus, Approximate entropy as a measure of system complexity, Proc. Natl. Acad. Sci. USA., 88 (1991), 2297-2301.  doi: 10.1073/pnas.88.6.2297. [14] F. Sapuppo, M. Bucolo, M. Intaglietta, L. Fortuna and P. Arena, A cellular nonlinear network: Real-time technology for the analysis of microfluidic phenomena in blood vessels, Nanotechnology, 17 (2006), S54. doi: 10.1088/0957-4484/17/4/009. [15] T. $\breve{S}$kovránek, I. Podlubny and I. Petrá$\breve{s}$, Modeling of the national economies in state-space: A fractional calculus approach, Economic Modelling, 29 (2012), 1322-1327. [16] H. Sun, Y. Zhang, D. Baleanu, W. Chen and Y. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlin. Sci. Numer. Simul., 64 (2018), 213-231.  doi: 10.1016/j.cnsns.2018.04.019. [17] I. Tejado, D. Valério and N. Valério, Fractional calculus in economic growth modeling. The Portuguese case, in ICFDA'14 International Conference on Fractional Differentiation and Its Applications, IEEE, 2014, 1–6. [18] B. Xin, W. Peng and Y. Kwon, A fractional-order difference Cournot duopoly game with long memory, preprint, arXiv: 1903.04305. doi: 10.1016/j.physa.2020.124993. [19] B. Xin and Y. Li, 0-1 test for chaos in a fractional order financial system with investment incentive, Abstract and Applied Analysis, 2013 (2013), 876298. doi: 10.1155/2013/876298. [20] B. Xin, W. Peng and L. Guerrini, A continuous time Bertrand duopoly game with fractional delay and conformable derivative: Modelling, discretization process, Hopf bifurcation and chaos, Frontiers in Physics, 7 (2019), 84. [21] C. Xu, M. Liao, P. Li, Q. Xiao and S. Yuan, Control strategy for a fractional-order chaotic financial model, Complexity, 2019 (2019), 2989204. doi: 10.1155/2019/2989204. [22] A. Yousefpour, H. Jahanshahi, J. M. Munoz-Pacheco, S. Bekiros and Z. Wei, A fractional-order hyper-chaotic economic system with transient chaos, Chaos, Solitons & Fractals, 130 (2020), 109400. doi: 10.1016/j.chaos.2019.109400.

show all references

##### References:
 [1] T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602-1611.  doi: 10.1016/j.camwa.2011.03.036. [2] A. Al-khedhairi, Differentiated Cournot duopoly game with fractional-order and its discretization, Engineering Computations, 36 (2019), 26. [3] A. Al-Khedhairi, Dynamics of a Cournot duopoly game with a generalized bounded rationality, Complexity, 2020 (2020), 8903183. [4] G. A. Anastassiou, Principles of delta fractional calculus on time scales and inequalities, Math. Comput. Model., 52 (2010), 556-566.  doi: 10.1016/j.mcm.2010.03.055. [5] S. S. Askar and A. Al-Khedhairi, Analysis of a four-firm competition based on a generalized bounded rationality and different mechanisms, Complexity, 2019 (2019), 6352796. doi: 10.1155/2019/6352796. [6] F. M. Atici and P. W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ. Spec. Ed. I, 3 (2009), 1-12.  doi: 10.14232/ejqtde.2009.4.3. [7] H. M. Baskonus, T. Mekkaoui, Z. Hammouch and H. Bulut, Active control of a chaotic fractional order economic system, Entropy, 17 (2015), 5771-5783.  doi: 10.3390/e17064255. [8] J. Cermak, I. Gyori and L. Nechvatal, On explicit stability conditions for a linear fractional difference system, Fractional Calculus and Applied Analysis, 18 (2015), 651-672. doi: 10.1515/fca-2015-0040. [9] G. A. Gottwald and I. Melbourne, On the implementation of the 0-1 test for chaos, SIAM J. Appl. Dyn. Syst., 8 (2009), 129-145.  doi: 10.1137/080718851. [10] N. Laskin, Fractional market dynamics, Physica A: Statist. Mech. Appl., 287 (2000), 482-492.  doi: 10.1016/S0378-4371(00)00387-3. [11] Y. Li, C. Sun, H. Ling, A. Lu and Y. Liu, Oligopolies price game in fractional order system, Chaos, Solitons & Fractals, 132 (2020), 109583. doi: 10.1016/j.chaos.2019.109583. [12] S. M. Pincus, Approximate entropy as a measure of system complexity, Proc. Natl. Acad. Sci. USA., 88 (1991), 2297–2301. doi: 10.1073/pnas.88.6.2297. [13] S. M. Pincus, Approximate entropy as a measure of system complexity, Proc. Natl. Acad. Sci. USA., 88 (1991), 2297-2301.  doi: 10.1073/pnas.88.6.2297. [14] F. Sapuppo, M. Bucolo, M. Intaglietta, L. Fortuna and P. Arena, A cellular nonlinear network: Real-time technology for the analysis of microfluidic phenomena in blood vessels, Nanotechnology, 17 (2006), S54. doi: 10.1088/0957-4484/17/4/009. [15] T. $\breve{S}$kovránek, I. Podlubny and I. Petrá$\breve{s}$, Modeling of the national economies in state-space: A fractional calculus approach, Economic Modelling, 29 (2012), 1322-1327. [16] H. Sun, Y. Zhang, D. Baleanu, W. Chen and Y. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlin. Sci. Numer. Simul., 64 (2018), 213-231.  doi: 10.1016/j.cnsns.2018.04.019. [17] I. Tejado, D. Valério and N. Valério, Fractional calculus in economic growth modeling. The Portuguese case, in ICFDA'14 International Conference on Fractional Differentiation and Its Applications, IEEE, 2014, 1–6. [18] B. Xin, W. Peng and Y. Kwon, A fractional-order difference Cournot duopoly game with long memory, preprint, arXiv: 1903.04305. doi: 10.1016/j.physa.2020.124993. [19] B. Xin and Y. Li, 0-1 test for chaos in a fractional order financial system with investment incentive, Abstract and Applied Analysis, 2013 (2013), 876298. doi: 10.1155/2013/876298. [20] B. Xin, W. Peng and L. Guerrini, A continuous time Bertrand duopoly game with fractional delay and conformable derivative: Modelling, discretization process, Hopf bifurcation and chaos, Frontiers in Physics, 7 (2019), 84. [21] C. Xu, M. Liao, P. Li, Q. Xiao and S. Yuan, Control strategy for a fractional-order chaotic financial model, Complexity, 2019 (2019), 2989204. doi: 10.1155/2019/2989204. [22] A. Yousefpour, H. Jahanshahi, J. M. Munoz-Pacheco, S. Bekiros and Z. Wei, A fractional-order hyper-chaotic economic system with transient chaos, Chaos, Solitons & Fractals, 130 (2020), 109400. doi: 10.1016/j.chaos.2019.109400.
The phase portraits of game (8) with parameter values $\alpha = 2, \, \varepsilon_1 = 1.042811791, \, \varepsilon _{2} = 1.1, \varepsilon _{3} = 1.1, \zeta_{1} = 0.4, \zeta_{2} = 0.8, \zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$ for different fractional order values: (a) $\nu = 1, (b)\, \nu = 0.9, (c) \, \nu = 0.865, (d)\, \nu = 0.81$
(a) Bifurcation diagram versus $\nu$ when $\alpha = 2, \varepsilon _{1} = 1.042811791, \varepsilon _{2} = 1.1, \varepsilon _{3} = 1.1, \zeta_{1} = 0.4, \zeta_{2} = 0.8, \zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$.(b) The maximum Lyapunov exponents with respect to $\nu$ corresponding to (a)
(a) Bifurcation diagram versus $\varepsilon _1$ with order $\nu = 0.985$ when $\alpha = 2, \varepsilon _{2} = 1.1, \varepsilon _{3} = 1.1, \zeta_{1} = 0.4, \zeta_{2} = 0.8, \zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$. (b) Bifurcation diagram versus $\varepsilon _{1}$ with order $\nu = 0.972.$
(a) Bifurcation diagram versus $\nu$ when $\alpha = 2, \varepsilon _{1} = 1.44, \varepsilon _{2} = 1.1, \varepsilon _{3} = 1.1, \zeta_{1} = 0.4, \zeta_{2} = 0.8, \zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$. (b) The maximum Lyapunov exponents with respect to $\nu$ corresponding to (a)
Chaotic attractor of the proposed game with $\nu =0.98$ and for $\alpha =2,% \varepsilon _{1}=1.44,\varepsilon _{2}=1.1,\varepsilon _{3}=1.1,\zeta_{1}=0.4,\zeta_{2}=0.8,\zeta_{3}=0.1,\gamma _{1}=0.07, \gamma _{2}=0.03,\gamma _{3}=0.4$.
Periodic attractor of the proposed game with $\nu = 0.975$ and for $\alpha = 2, \varepsilon _{1} = 1.44,\varepsilon _{2} = 1.1,\varepsilon _{3} = 1.1,\zeta_{1} = 0.4,\zeta_{2} = 0.8,\zeta_{3} = 0.1,\gamma _{1} = 0.07, \gamma _{2} = 0.03,\gamma _{3} = 0.4$
Chaotic attractor of the proposed game with $\nu = 0.96$ and for $\alpha = 2, \varepsilon _{1} = 1.44, \varepsilon _{2} = 1.1, \varepsilon _{3} = 1.1, \zeta_{1} = 0.4, \zeta_{2} = 0.8, \zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$
(a) Bifurcation diagram versus $\varepsilon_1$ with order $\nu = 1$ when $\alpha = 1, \varepsilon _{2} = 0.9, \varepsilon _{3} = 0.9, \zeta_{1} = 0.4, \zeta_{2} = 0.8, \zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$ (b) Bifurcation diagram versus $\varepsilon _{1}$ with order $\nu = 0.7635.$
Periodic attractor of the proposed game with $\nu = 0.7635$ for $\varepsilon_1 = 2.6$ and $\alpha = 1, \varepsilon _{2} = 0.9,\varepsilon _{3} = 0.9,\zeta_{1} = 0.4,\zeta_{2} = 0.8,\zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$
Chaotic attractor of the proposed game with $\nu = 0.7635$ for $\varepsilon_1 = 2.9$ and $\alpha = 1, \varepsilon _{2} = 0.9,\varepsilon _{3} = 0.9,\zeta_{1} = 0.4,\zeta_{2} = 0.8,\zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$
0-1 test: regular dynamics of the translation components $(p, q)$ of the Cournot game (8) for $\alpha = 2$, $\varepsilon _{2} = 1.1, \varepsilon _{3} = 1.1, \zeta_{1} = 0.4, \zeta_{2} = 0.8, \zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$ with fractional order $\nu = 0.865$
0-1 test: regular dynamics of the translation components $(p, q)$ of the Cournot game (8) for $\alpha = 2, \varepsilon _{2} = 1.1, \varepsilon _{3} = 1.1, \zeta_{1} = 0.4, \zeta_{2} = 0.8, \zeta_{3} = 0.1, \gamma _{1} = 0.07, \gamma _{2} = 0.03, \gamma _{3} = 0.4$ with fractional order $\nu = 0.7635$
ApEn of the game model (8) vesrus $\nu$
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