On the relationship between the open-loop Nash equilibrium in LQ-games and the inertia of a matrix

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

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Abstract

In this paper we consider the location of the eigenvalues of the composite matrix ( -A S1 S2 ) ( Q1 At 0 ) ( Q2 0 At ) , where the matrices Si and Qi are assumed to be semi-positive definite. Two interesting observations, which are not or only partially mentioned in literature before, challenge this study. The first observation is that this matrix appears naturally in a both necessary and sufficient condition for the existence of a unique open-loop Nash solution in the 2-player linear-quadratic dynamic game and, more in particular, its inertia play an important role in the analysis of the convergence of the associated state in this game. The second observation is that from the eigenvalue and eigenstructure of this matrix all solutions for the algebraic Riccati equations corresponding with the above mentioned dynamic game can be directly calculated and, moreover, also the eigenvalues of the associated closed-loop system. Simulation experiments suggest that the composite matrix will have at least n eigenvalues (here n is the state dimension of the system) with a positive real part. Unfortunately, it turns out that this property of the inertia of this matrix in general does not hold. Some specific cases for which the property does hold are discussed.
Original languageEnglish
PublisherUnknown Publisher
Number of pages14
VolumeFEW 672
Publication statusPublished - 1994

Publication series

NameResearch memorandum / Tilburg University, Department of Economics
VolumeFEW 672

Fingerprint

Nash Equilibrium
Inertia
Game
Eigenvalue
Dynamic Games
Composite
Algebraic Riccati Equation
Positive semidefinite
Closed-loop System
Simulation Experiment
Relationships
Necessary Conditions
Sufficient Conditions
Observation

Keywords

  • Game Theory
  • Nash Equilibrium
  • game theory

Cite this

Engwerda, J. C., & Weeren, A. J. T. M. (1994). On the relationship between the open-loop Nash equilibrium in LQ-games and the inertia of a matrix. (Research memorandum / Tilburg University, Department of Economics; Vol. FEW 672). Unknown Publisher.
Engwerda, J.C. ; Weeren, A.J.T.M. / On the relationship between the open-loop Nash equilibrium in LQ-games and the inertia of a matrix. Unknown Publisher, 1994. 14 p. (Research memorandum / Tilburg University, Department of Economics).
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Engwerda, JC & Weeren, AJTM 1994, On the relationship between the open-loop Nash equilibrium in LQ-games and the inertia of a matrix. Research memorandum / Tilburg University, Department of Economics, vol. FEW 672, vol. FEW 672, Unknown Publisher.

On the relationship between the open-loop Nash equilibrium in LQ-games and the inertia of a matrix. / Engwerda, J.C.; Weeren, A.J.T.M.

Unknown Publisher, 1994. 14 p. (Research memorandum / Tilburg University, Department of Economics; Vol. FEW 672).

Research output: Book/ReportReportProfessional

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PY - 1994

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N2 - In this paper we consider the location of the eigenvalues of the composite matrix ( -A S1 S2 ) ( Q1 At 0 ) ( Q2 0 At ) , where the matrices Si and Qi are assumed to be semi-positive definite. Two interesting observations, which are not or only partially mentioned in literature before, challenge this study. The first observation is that this matrix appears naturally in a both necessary and sufficient condition for the existence of a unique open-loop Nash solution in the 2-player linear-quadratic dynamic game and, more in particular, its inertia play an important role in the analysis of the convergence of the associated state in this game. The second observation is that from the eigenvalue and eigenstructure of this matrix all solutions for the algebraic Riccati equations corresponding with the above mentioned dynamic game can be directly calculated and, moreover, also the eigenvalues of the associated closed-loop system. Simulation experiments suggest that the composite matrix will have at least n eigenvalues (here n is the state dimension of the system) with a positive real part. Unfortunately, it turns out that this property of the inertia of this matrix in general does not hold. Some specific cases for which the property does hold are discussed.

AB - In this paper we consider the location of the eigenvalues of the composite matrix ( -A S1 S2 ) ( Q1 At 0 ) ( Q2 0 At ) , where the matrices Si and Qi are assumed to be semi-positive definite. Two interesting observations, which are not or only partially mentioned in literature before, challenge this study. The first observation is that this matrix appears naturally in a both necessary and sufficient condition for the existence of a unique open-loop Nash solution in the 2-player linear-quadratic dynamic game and, more in particular, its inertia play an important role in the analysis of the convergence of the associated state in this game. The second observation is that from the eigenvalue and eigenstructure of this matrix all solutions for the algebraic Riccati equations corresponding with the above mentioned dynamic game can be directly calculated and, moreover, also the eigenvalues of the associated closed-loop system. Simulation experiments suggest that the composite matrix will have at least n eigenvalues (here n is the state dimension of the system) with a positive real part. Unfortunately, it turns out that this property of the inertia of this matrix in general does not hold. Some specific cases for which the property does hold are discussed.

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Engwerda JC, Weeren AJTM. On the relationship between the open-loop Nash equilibrium in LQ-games and the inertia of a matrix. Unknown Publisher, 1994. 14 p. (Research memorandum / Tilburg University, Department of Economics).