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A numerical iterative method for solving two-point BVPs in infinite-horizon nonzero-sum differential games: Economic applications

Author

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  • Nikooeinejad, Z.
  • Heydari, M.
  • Loghmani, G.B.

Abstract

In this work, the Nash equilibrium solution of nonlinear infinite-horizon nonzero-sum differential games with open-loop information is investigated numerically. For this class of games, some difficulties are involved in finding an open-loop Nash equilibrium. For instance, we must solve a nonlinear differential equations system with split boundary conditions, namely the two-point boundary value problem (TPBVP), in which some of the boundary conditions are specified at the initial time and some at the infinite final time. In the current study, we provide a combined numerical algorithm based on a new set of basis functions on the half-line, called the exponential Chelyshkov functions (ECFs), and quasilinearization (QL) method to solve TPBVPs. In the first step, we reduce the nonlinear TPBVP to a sequence of linear differential equations by using the QL method. Although we have now a linearized system, another difficulty is finding the approximate solution of this linear system such that the transversality conditions at the infinite final time are satisfied. So, in the second step, we apply a collocation method based on the ECFs to solve the obtained linear system in the semi-infinite domain. The convergence of the proposed method is discussed in detail. To confirm the validity and efficiency of the proposed scheme, we compute the approximate solution of TPBVP as well as the open-loop Nash equilibrium for four applications of differential games in economics and management science.

Suggested Citation

  • Nikooeinejad, Z. & Heydari, M. & Loghmani, G.B., 2022. "A numerical iterative method for solving two-point BVPs in infinite-horizon nonzero-sum differential games: Economic applications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 404-427.
  • Handle: RePEc:eee:matcom:v:200:y:2022:i:c:p:404-427
    DOI: 10.1016/j.matcom.2022.04.022
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    References listed on IDEAS

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    1. J.C. Engwerda & B. Aarle & J.E.J. Plasmans, 1999. "The (in)finite horizon open‐loop Nash LQ game:An application to EMU," Annals of Operations Research, Springer, vol. 88(0), pages 251-273, January.
    2. Engwerda, Jacob C., 1998. "On the open-loop Nash equilibrium in LQ-games," Journal of Economic Dynamics and Control, Elsevier, vol. 22(5), pages 729-762, May.
    3. Fershtman, Chaim & Kamien, Morton I, 1987. "Dynamic Duopolistic Competition with Sticky Prices," Econometrica, Econometric Society, vol. 55(5), pages 1151-1164, September.
    4. Engwerda, J.C. & van den Broek, W.A. & Schumacher, J.M., 2000. "Feedback Nash equilibria in uncertain infinite time horizon differential games," Other publications TiSEM c431993d-ee67-4a93-9e2d-f, Tilburg University, School of Economics and Management.
    5. Oğuz, Cem & Sezer, Mehmet, 2015. "Chelyshkov collocation method for a class of mixed functional integro-differential equations," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 943-954.
    6. Jacob Engwerda & Davoud Mahmoudinia & Rahim Dalali Isfahani, 2016. "Government and Central Bank Interaction under Uncertainty: A Differential Games Approach," Iranian Economic Review (IER), Faculty of Economics,University of Tehran.Tehran,Iran, vol. 20(2), pages 225-259, Spring.
    7. Umer Saeed & Mujeeb ur Rehman, 2014. "Wavelet-Galerkin Quasilinearization Method for Nonlinear Boundary Value Problems," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-10, June.
    8. S. Hristova & A. Golev & K. Stefanova, 2012. "Quasilinearization of the Initial Value Problem for Difference Equations with “Maxima”," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-17, September.
    9. Dockner,Engelbert J. & Jorgensen,Steffen & Long,Ngo Van & Sorger,Gerhard, 2000. "Differential Games in Economics and Management Science," Cambridge Books, Cambridge University Press, number 9780521637329, October.
    10. Engwerda, J.C., 2000. "Feedback Nash equilibria in the scalar infinite horizon LQ-Game," Other publications TiSEM 58ccf964-4ca1-4d67-9a68-a, Tilburg University, School of Economics and Management.
    11. Sorger, Gerhard, 1989. "Competitive dynamic advertising : A modification of the Case game," Journal of Economic Dynamics and Control, Elsevier, vol. 13(1), pages 55-80, January.
    12. Tabellini, Guido, 1986. "Money, debt and deficits in a dynamic game," Journal of Economic Dynamics and Control, Elsevier, vol. 10(4), pages 427-442, December.
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