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The Solution Set of the n-Player Scalar Feedback Nash Algebraic Riccati Equations

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  • Engwerda, J.C.

    (Tilburg University, Center For Economic Research)

Abstract

In this paper we analyse the set of scalar algebraic Riccati equations (ARE) that play an important role in finding feedback Nash equilibria of the scalar N-player linear-quadratic differential game. We show that in general there exist maximal 2N - 1 solutions of the (ARE) that give rise to a Nash equilibrium. In particular we analyse the number of equilibria as a function of the state-feedback parameter and present both necessary and sufficient conditions for existence of a unique solution of the (ARE). Furthermore, we derive conditions under which the set of state-feedback parameters for which there is a unique solution grows with the number of players in the game.
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Suggested Citation

  • Engwerda, J.C., 1999. "The Solution Set of the n-Player Scalar Feedback Nash Algebraic Riccati Equations," Discussion Paper 1999-90, Tilburg University, Center for Economic Research.
    • Handle: RePEc:tiu:tiucen:63f19390-d8dd-4c84-9b96-7161ab804989
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    File URL: https://pure.uvt.nl/ws/portalfiles/portal/533176/90.pdf
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    1. Weeren, A.J.T.M. & Schumacher, J.M. & Engwerda, J.C., 1994. "Asymptotic analysis of Nash equilibria in nonzero-sum linear-quadratic differential games : The two player case," Research Memorandum FEW 634, Tilburg University, School of Economics and Management.
    2. 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.
    3. 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.
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    Cited by:

    1. Engwerda, J.C., 2013. "A Numerical Algorithm to find All Scalar Feedback Nash Equilibria," Discussion Paper 2013-050, Tilburg University, Center for Economic Research.
    2. Jacob Engwerda, 2017. "A Numerical Algorithm to Calculate the Unique Feedback Nash Equilibrium in a Large Scalar LQ Differential Game," Dynamic Games and Applications, Springer, vol. 7(4), pages 635-656, December.
    3. Acocella, Nicola & Di Bartolomeo, Giovanni, 2007. "Towards a new theory of economic policy: Continuity and innovation," MPRA Paper 4419, University Library of Munich, Germany.
    4. Engwerda, J.C., 2004. "A numerical algorithm to find soft-constrained Nash equilibria in scalar LQ-games," Other publications TiSEM 7a3232f4-ef03-4cc7-a438-e, Tilburg University, School of Economics and Management.
    5. Tamer Başar & Quanyan Zhu, 2011. "Prices of Anarchy, Information, and Cooperation in Differential Games," Dynamic Games and Applications, Springer, vol. 1(1), pages 50-73, March.

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