The open-loop Nash equilibrium in LQ-games revisited

J.C. Engwerda, A.J.T.M. Weeren

Research output: Book/ReportReportProfessional

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Abstract

In this paper we reconsider the conditions under which the finite-planning-horizon linear- quadratic differential game has an open-loop Nash equilibrium solution. Both necessary and sufficient conditions are presented for the existence of a unique solution in terms of an invertibility condition on a matrix. Moreover, we show that the often encountered solvability conditions stated in terms of Riccati equations are in general not correct. In an example we show that existence of a solution of the associated Riccati-type differential equations may fail to exist whereas an open-loop Nash equilibrium solution exists. The scalar case is studied in more detail, and we show that solvability of the associated Riccati equations is in that case both necessary and sufficient. Furthermore we consider convergence properties of the open-loop Nash equilibrium solution if the planning horizon is extended to infinity. To study this aspect we consider the existence of real solutions of the associated algebraic Riccati equation in detail and show how all solutions can be easily calculated from the eigenstructure of a matrix.
Original languageEnglish
PublisherUnknown Publisher
Number of pages19
Volume9551
Publication statusPublished - 1995

Publication series

NameDiscussion Papers / CentER for Economic Research
Volume9551

Fingerprint

Equilibrium Solution
Nash Equilibrium
Riccati Equation
Game
Horizon
Planning
Quadratic Differentials
Solvability Conditions
Algebraic Riccati Equation
Invertibility
Differential Games
Unique Solution
Convergence Properties
Solvability
Infinity
Scalar
Sufficient
Differential equation
Necessary Conditions
Necessary

Keywords

  • Nash Equilibrium
  • Game Theory
  • game theory

Cite this

Engwerda, J. C., & Weeren, A. J. T. M. (1995). The open-loop Nash equilibrium in LQ-games revisited. (Discussion Papers / CentER for Economic Research; Vol. 9551). Unknown Publisher.
Engwerda, J.C. ; Weeren, A.J.T.M. / The open-loop Nash equilibrium in LQ-games revisited. Unknown Publisher, 1995. 19 p. (Discussion Papers / CentER for Economic Research).
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Engwerda, JC & Weeren, AJTM 1995, The open-loop Nash equilibrium in LQ-games revisited. Discussion Papers / CentER for Economic Research, vol. 9551, vol. 9551, Unknown Publisher.

The open-loop Nash equilibrium in LQ-games revisited. / Engwerda, J.C.; Weeren, A.J.T.M.

Unknown Publisher, 1995. 19 p. (Discussion Papers / CentER for Economic Research; Vol. 9551).

Research output: Book/ReportReportProfessional

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AB - In this paper we reconsider the conditions under which the finite-planning-horizon linear- quadratic differential game has an open-loop Nash equilibrium solution. Both necessary and sufficient conditions are presented for the existence of a unique solution in terms of an invertibility condition on a matrix. Moreover, we show that the often encountered solvability conditions stated in terms of Riccati equations are in general not correct. In an example we show that existence of a solution of the associated Riccati-type differential equations may fail to exist whereas an open-loop Nash equilibrium solution exists. The scalar case is studied in more detail, and we show that solvability of the associated Riccati equations is in that case both necessary and sufficient. Furthermore we consider convergence properties of the open-loop Nash equilibrium solution if the planning horizon is extended to infinity. To study this aspect we consider the existence of real solutions of the associated algebraic Riccati equation in detail and show how all solutions can be easily calculated from the eigenstructure of a matrix.

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Engwerda JC, Weeren AJTM. The open-loop Nash equilibrium in LQ-games revisited. Unknown Publisher, 1995. 19 p. (Discussion Papers / CentER for Economic Research).