DOI QR코드

DOI QR Code

OPTIMAL STRATEGIES IN BIOECONOMIC DIFFERENTIAL GAMES: INSIGHTS FROM CHEBYSHEV TAU METHOD

  • Shahd H. Alkharaz (Department of Mathematics, Faculty of Science, Ain Shams University) ;
  • Essam El-Siedy (Pure Mathematics, Mathematics Department, Faculty of Science, Ain Shams University) ;
  • Eliwa M. Roushdy (Department of Basic and Applied Sciences, Arab Academy for Science, Technology and Maritime Transport) ;
  • Muner M. Abou Hasan (School of Mathematics and data science, Emirates aviation university)
  • Received : 2023.10.08
  • Accepted : 2023.11.26
  • Published : 2024.06.15

Abstract

In the realm of differential games and bioeconomic modeling, where intricate systems and multifaceted interactions abound, we explore the precision and efficiency of the Chebyshev Tau method (CTM). We begin with the Weierstrass Approximation Theorem, employing Chebyshev polynomials to pave the way for solving intricate bioeconomic differential games. Our case study revolves around a three-player bioeconomic differential game, unveiling a unique open-loop Nash equilibrium using Hamiltonians and the FilippovCesari existence theorem. We then transition to numerical implementation, employing CTM to resolve a Three-Point Boundary Value Problem (TPBVP) with varying degrees of approximation.

Keywords

References

  1. A. Bressan, Bifurcation analysis of a non-cooperative differential game with one weak player, J. Dif. Equ., 248 (2010), 1297-1314.  https://doi.org/10.1016/j.jde.2009.11.025
  2. A. Bressan, Noncooperative differential games. A tutorial. Milan J. Math., 79 (2011), 357-427.  https://doi.org/10.1007/s00032-011-0163-6
  3. C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer: New York, NY, USA, 2006. 
  4. D.A. Carlson and G. Leitmann, An extension of the coordinate transformation method for open-loop Nash equilibria, J. Optim. Theory Appl., 123 (2004), 27-47.  https://doi.org/10.1023/B:JOTA.0000043990.32923.ac
  5. L. Cesari, Optimization-Theory and Applications: Problems with Ordinary Differential Equations. Springer: New York, NY, USA, 1983. 
  6. E. Dockner, S. Jorgensen, N. Van Long and G. Sorger, Differential Games in Economics and Management Science. Cambridge University Press: Cambridge, UK, 2000. 
  7. E.H. Doha, W.M. Abd-Elhameed and A.H. Bhrawy, Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2nth-order linear differential equations, Appl. Math. Model., 33 (2009), 1982-1996.  https://doi.org/10.1016/j.apm.2008.05.005
  8. E.H. Doha, A.H. Bhrawy and R.M. Hafez, On shifted Jacobi spectral method for high-order multi-point boundary value problems, Commu.. Nonlinear Sci. Numer. Simul., 17 (2012), 3802-3810.  https://doi.org/10.1016/j.cnsns.2012.02.027
  9. J.C. Engwerda, LQ Dynamic Optimization and Differential Games, John Wiley and Sons: Hoboken, NJ, USA, 2005. 
  10. J.C. Engwerda, On the open-loop Nash equilibrium in LQ games, J. Econom. Dynam. Control, 22 (1998), 729-762.  https://doi.org/10.1016/S0165-1889(97)00084-5
  11. J.C. Engwerda, Feedback Nash equilibria in the scalar infinite horizon LQ game, Automatica, 36(2000), 135-139.  https://doi.org/10.1016/S0005-1098(99)00119-3
  12. G. Erickson, "Dynamic Models of Advertising Competition." Kluwer: Boston, MA, USA, 2003. 
  13. L. Grosset, A note on open loop Nash equilibrium in linear-state differential games, Appl. Math. Sci., 8 (2014), 7239-7248. 
  14. B.Y. Guo, Spectral Methods and Their Applications, World Scientific: Singapore, 1998. 
  15. S. Jafari and H. Navidi, A game-theoretic approach for modeling competitive diffusion over social networks, Games, 9 (2018). 
  16. M. Jimenez-Lizarraga, M. Basin, V. Rodriguez and P. Rodriguez, Open-loop Nash equilibrium in polynomial differential games via state-dependent Riccati equation, Automatica, 53 (2015), 155-163.  https://doi.org/10.1016/j.automatica.2014.12.035
  17. G. Kossiorisa, M. Plexousakis, A. Xepapadeas, A. de Zeeuwe and K.G. Maler, Feedback Nash equilibria for non-linear differential games in pollution control, J. Econom. Dynam. Control, 32 (2008), 1312-1331.  https://doi.org/10.1016/j.jedc.2007.05.008
  18. R. Moosavi Mohseni, Mathematical Analysis of the Chaotic Behavior in Monetary Policy Games, Ph.D. Thesis, Auckland University of Technology, Auckland, New Zealand, 2019. 
  19. Z. Nikooeinejad, A. Delavarkhalafi and M. Heydari, A numerical solution of open-loop Nash equilibrium in nonlinear differential games based on Chebyshev pseudospectral method, J. Comput. Appl. Math., 300 (2016), 369-384.  https://doi.org/10.1016/j.cam.2016.01.019
  20. G. Sorger, Competitive dynamic advertising: A modification of the case game, J. Econom. Dynam. Control, 13 (1989), 55-80.  https://doi.org/10.1016/0165-1889(89)90011-0
  21. A. Starr and Y. Ho, Further properties of nonzero-sum differential games, J. Optim. Theory Appl., 3 (1969), 207-219.  https://doi.org/10.1007/BF00926523
  22. A. Starr and Y. Ho, Nonzero-sum differential games, J. Optim. Theory Appl., 3 (1969), 184-206.  https://doi.org/10.1007/BF00929443
  23. NH. Sweilam and MM. Abou Hasan, An Improved Method for Nonlinear Variable-Order LevyFeller AdvectionDispersion Equation, Bull. Malaysian Math. Sci. Soc., 42(6) (2019), 3021-3046.  https://doi.org/10.1007/s40840-018-0644-7
  24. NH. Sweilam and MM. Abou Hasan. Efficient Method for Fractional Levy-Feller Advection-Dispersion Equation Using Jacobi Polynomials, Progress Fractional Dif. Appl., 6(2) (2018), doi.org/10.48550/arXiv.1803.03143. 
  25. NH. Sweilam, TA. Assiri and MM. Abou Hasan, Optimal control problem of variable-order delay system of advertising procedure: Numerical treatment, Discrete Conti. Dyn. Syst., 15(5) (2022), 1247. 
  26. NH. Sweilam, AF. Ghaleb, MS. Abou-Dina and MM. Abou Hasan, Numerical solution to a one-dimensional nonlinear problem of heat wave propagation in a rigid thermal conducting slab, Indian J. Phy., 96(1), 223-232. 
  27. D. Yeung, and L. Petrosjan, Cooperative Stochastic Differential Games, Springer: Berlin/Heidelberg, Germany, 2005. 
  28. H. Zhang, Q. Wei and D. Liu, An iterative dynamic programming method for solving a class of nonlinear zero-sum differential games, Automatica, 47 (2011), 207-214. https://doi.org/10.1016/j.automatica.2010.10.033