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


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

    (Tilburg University, Faculty of Economics and Business Administration)


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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Bibliographic Info

Paper provided by Tilburg University, Faculty of Economics and Business Administration in its series Research Memorandum with number 672.

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Date of creation: 1994
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Handle: RePEc:dgr:kubrem:1994672

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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 634, Tilburg University, Faculty of Economics and Business Administration.
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