### Abstract

Original language | English |
---|---|

Publisher | Unknown Publisher |

Number of pages | 19 |

Volume | 9551 |

Publication status | Published - 1995 |

### Publication series

Name | Discussion Papers / CentER for Economic Research |
---|---|

Volume | 9551 |

### Fingerprint

### Keywords

- Nash Equilibrium
- Game Theory
- game theory

### Cite this

*The open-loop Nash equilibrium in LQ-games revisited*. (Discussion Papers / CentER for Economic Research; Vol. 9551). Unknown Publisher.

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*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.

Research output: Book/Report › Report › Professional

TY - BOOK

T1 - The open-loop Nash equilibrium in LQ-games revisited

AU - Engwerda, J.C.

AU - Weeren, A.J.T.M.

N1 - Pagination: 19

PY - 1995

Y1 - 1995

N2 - 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.

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.

KW - Nash Equilibrium

KW - Game Theory

KW - game theory

M3 - Report

VL - 9551

T3 - Discussion Papers / CentER for Economic Research

BT - The open-loop Nash equilibrium in LQ-games revisited

PB - Unknown Publisher

ER -