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An Extension of the Coordinate Transformation Method for Open-Loop Nash Equilibria

Author

Listed:
  • D. A. Carlson

    (University of Toledo)

  • G. Leitmann

    (University of California)

Abstract

In Leitmann (Ref. 1), a coordinate transformation method was introduced to obtain global solutions for free problems in the calculus of variations. This direct method was extended and broadened in Carlson (Ref. 2) and later in Leitmann (Ref. 3). The applicability of the original work of Leitmann (Ref. 1) was further developed in Dockner and Leitmann (Ref. 4) to include the class of open-loop dynamic games. In the present work, we improve the results of Ref. 4 in two directions. First, we enlarge the class of open-loop dynamic games to permit coupling among the dynamic equations via the states of the players; second, we incorporate the modifications given in Refs. 2 and 3. Our results greatly increase the applicability of this method. An example arising from the harvesting of a renewable resource is presented to illustrate the utility of our results.

Suggested Citation

  • D. A. Carlson & G. Leitmann, 2004. "An Extension of the Coordinate Transformation Method for Open-Loop Nash Equilibria," Journal of Optimization Theory and Applications, Springer, vol. 123(1), pages 27-47, October.
    • Handle: RePEc:spr:joptap:v:123:y:2004:i:1:d:10.1023_b:jota.0000043990.32923.ac
    DOI: 10.1023/B:JOTA.0000043990.32923.ac
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    References listed on IDEAS

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    1. D.A. Carlson, 2002. "An Observation on Two Methods of Obtaining Solutions to Variational Problems," Journal of Optimization Theory and Applications, Springer, vol. 114(2), pages 345-361, August.
    2. E. J. Dockner & G. Leitmann, 2001. "Coordinate Transformations and Derivation of Open-Loop Nash Equilibria," Journal of Optimization Theory and Applications, Springer, vol. 110(1), pages 1-15, July.
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    Cited by:

    1. S. G. Ji & S. Y. Shi, 2006. "Periodic Solutions for a Class of Second-Order Ordinary Differential Equations," Journal of Optimization Theory and Applications, Springer, vol. 130(1), pages 125-137, July.
    2. Jiangjing Zhou & Anna Tur & Ovanes Petrosian & Hongwei Gao, 2021. "Transferable Utility Cooperative Differential Games with Continuous Updating Using Pontryagin Maximum Principle," Mathematics, MDPI, vol. 9(2), pages 1-22, January.
    3. Mojtaba Dehghan Banadaki & Hamidreza Navidi, 2020. "Numerical Solution of Open-Loop Nash Differential Games Based on the Legendre Tau Method," Games, MDPI, vol. 11(3), pages 1-11, July.

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