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Asymptotic Analysis of Linear Feedback Nash Equilibria in Nonzero-Sum Linear-Quadratic Differential Games

Author

Listed:
  • A. J. T. M. Weeren

    (University of Antwerp))

  • J. M. Schumacher

    (Tilburg University)

  • J. C. Engwerda

    (Tilburg University)

Abstract

In this paper, we discuss nonzero-sum linear-quadratic differential games. For this kind of games, the Nash equilibria for different kinds of information structures were first studied by Starr and Ho. Most of the literature on the topic of nonzero-sum linear-quadratic differential games is concerned with games of fixed, finite duration; i.e., games are studied over a finite time horizon t f. In this paper, we study the behavior of feedback Nash equilibria for t f→∞. In the case of memoryless perfect-state information, we study the so-called feedback Nash equilibrium. Contrary to the open-loop case, we note that the coupled Riccati equations for the feedback Nash equilibrium are inherently nonlinear. Therefore, we limit the dynamic analysis to the scalar case. For the special case that all parameters are scalar, a detailed dynamical analysis is given for the quadratic system of coupled Riccati equations. We show that the asymptotic behavior of the solutions of the Riccati equations depends strongly on the specified terminal values. Finally, we show that, although the feedback Nash equilibrium over any fixed finite horizon is generically unique, there can exist several different feedback Nash equilibria in stationary strategies for the infinite-horizon problem, even when we restrict our attention to Nash equilibria that are stable in the dynamical sense.

Suggested Citation

  • A. J. T. M. Weeren & J. M. Schumacher & J. C. Engwerda, 1999. "Asymptotic Analysis of Linear Feedback Nash Equilibria in Nonzero-Sum Linear-Quadratic Differential Games," Journal of Optimization Theory and Applications, Springer, vol. 101(3), pages 693-722, June.
    • Handle: RePEc:spr:joptap:v:101:y:1999:i:3:d:10.1023_a:1021798322597
    DOI: 10.1023/A:1021798322597
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    References listed on IDEAS

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    1. Engwerda, J.C. & Weeren, A.J.T.M., 1995. "The open-loop Nash equilibrium in LQ-games revisited," Other publications TiSEM 1792b29c-db37-431e-b900-8, Tilburg University, School of Economics and Management.
    2. 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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    Cited by:

    1. Engwerda, J.C., 1999. "The Solution Set of the n-Player Scalar Feedback Nash Algebraic Riccati Equations," Other publications TiSEM 63f19390-d8dd-4c84-9b96-7, Tilburg University, School of Economics and Management.
    2. Engwerda, J.C., 1998. "On the Scalar Feedback Nash Equilibria in the Infinite Horizon LQ-Game," Discussion Paper 1998-112, Tilburg University, Center for Economic Research.
    3. Engwerda, J.C., 2013. "A Numerical Algorithm to find All Scalar Feedback Nash Equilibria," Discussion Paper 2013-050, Tilburg University, Center for Economic Research.
    4. Engwerda, Jacob & van Aarle, Bas & Plasmans, Joseph & Weeren, Arie, 2013. "Debt stabilization games in the presence of risk premia," Journal of Economic Dynamics and Control, Elsevier, vol. 37(12), pages 2525-2546.
    5. Engwerda, J.C. & Salmah, Y., 2010. "Necessary and Sufficient Conditions for Feedback Nash Equilibria for the Affine Quadratic Differential," Other publications TiSEM 4be56827-dca1-42c3-8872-6, Tilburg University, School of Economics and Management.
    6. J. C. Engwerda & Salmah, 2013. "Necessary and Sufficient Conditions for Feedback Nash Equilibria for the Affine-Quadratic Differential Game," Journal of Optimization Theory and Applications, Springer, vol. 157(2), pages 552-563, May.
    7. P. Cartigny & P. Michel, 2003. "On the Selection of One Feedback Nash Equilibrium in Discounted Linear-Quadratic Games," Journal of Optimization Theory and Applications, Springer, vol. 117(2), pages 231-243, May.
    8. Eigruber, Markus & Wirl, Franz, 2024. "Market equilibrium strategies under learning by doing and spillovers," Energy Economics, Elsevier, vol. 131(C).
    9. Bas Van Aarle & Jacob Engwerda & Joseph Plasmans & Arie Weeren, 2001. "Macroeconomic Policy Interaction under EMU: A Dynamic Game Approach," Open Economies Review, Springer, vol. 12(1), pages 29-60, January.
    10. van den Broek, W.A. & Engwerda, J.C. & Schumacher, J.M., 2003. "An equivalence result in linear-quadratic theory," Other publications TiSEM d65171ce-101d-4204-a1ec-f, Tilburg University, School of Economics and Management.
    11. 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.
    12. Alberto Bressan & Khai T. Nguyen, 2018. "Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games," Dynamic Games and Applications, Springer, vol. 8(1), pages 42-78, March.

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