Robust open-loop Nash equilibria in the non-cooperative LQ game revisited

J.C. Engwerda

Research output: Chapter in Book/Report/Conference proceedingConference contributionScientificpeer-review

Abstract

This paper reconsiders existence of worst-case Nash equilibria in noncooperative multi-player differential games, this, within an open-loop information structure. We show that these equilibria can be obtained by determining the open-loop Nash equilibria of an associated differential game with an additional initial state constraint. For the special case of linear-quadratic differential games, we derive both necessary and sufficient conditions for solvability of the finite planning horizon problem. In particular, we demonstrate that, unlike in the standard linear-quadratic differential game setting, uniqueness of equilibria may fail to hold. A both necessary and sufficient condition under which there is a unique equilibrium is provided. A sufficient existence condition for a unique equilibrium is derived in terms of a Riccati differential equation. Consequences for control policies are demonstrated in a simple debt stabilization game.
Original languageEnglish
Title of host publicationProceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014)
Place of PublicationGroningen
PublisherUniversity of Groningen
Pages200-207
ISBN (Print)9789036763219
DOIs
Publication statusPublished - Jul 2014
Event21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014) - University of Groningen, Groningen, Netherlands
Duration: 7 Jul 201411 Jul 2014

Conference

Conference21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014)
CountryNetherlands
CityGroningen
Period7/07/1411/07/14

Fingerprint

Open-loop Nash equilibrium
Non-cooperative game
Linear quadratic differential games
Differential games
Nash equilibrium
Differential equations
Solvability
Debt
State constraints
Information structure
Uniqueness
Stabilization
Planning

Keywords

  • robust control
  • noncooperative differential games
  • linear optimal control
  • Riccati equations

Cite this

Engwerda, J. C. (2014). Robust open-loop Nash equilibria in the non-cooperative LQ game revisited. In Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014) (pp. 200-207). Groningen: University of Groningen. https://doi.org/10.1002/oca.2290
Engwerda, J.C. / Robust open-loop Nash equilibria in the non-cooperative LQ game revisited. Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014). Groningen : University of Groningen, 2014. pp. 200-207
@inproceedings{c81cf4fe73c04b4fbac551b052da70c9,
title = "Robust open-loop Nash equilibria in the non-cooperative LQ game revisited",
abstract = "This paper reconsiders existence of worst-case Nash equilibria in noncooperative multi-player differential games, this, within an open-loop information structure. We show that these equilibria can be obtained by determining the open-loop Nash equilibria of an associated differential game with an additional initial state constraint. For the special case of linear-quadratic differential games, we derive both necessary and sufficient conditions for solvability of the finite planning horizon problem. In particular, we demonstrate that, unlike in the standard linear-quadratic differential game setting, uniqueness of equilibria may fail to hold. A both necessary and sufficient condition under which there is a unique equilibrium is provided. A sufficient existence condition for a unique equilibrium is derived in terms of a Riccati differential equation. Consequences for control policies are demonstrated in a simple debt stabilization game.",
keywords = "robust control, noncooperative differential games, linear optimal control, Riccati equations",
author = "J.C. Engwerda",
year = "2014",
month = "7",
doi = "10.1002/oca.2290",
language = "English",
isbn = "9789036763219",
pages = "200--207",
booktitle = "Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014)",
publisher = "University of Groningen",

}

Engwerda, JC 2014, Robust open-loop Nash equilibria in the non-cooperative LQ game revisited. in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014). University of Groningen, Groningen, pp. 200-207, 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014), Groningen, Netherlands, 7/07/14. https://doi.org/10.1002/oca.2290

Robust open-loop Nash equilibria in the non-cooperative LQ game revisited. / Engwerda, J.C.

Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014). Groningen : University of Groningen, 2014. p. 200-207.

Research output: Chapter in Book/Report/Conference proceedingConference contributionScientificpeer-review

TY - GEN

T1 - Robust open-loop Nash equilibria in the non-cooperative LQ game revisited

AU - Engwerda, J.C.

PY - 2014/7

Y1 - 2014/7

N2 - This paper reconsiders existence of worst-case Nash equilibria in noncooperative multi-player differential games, this, within an open-loop information structure. We show that these equilibria can be obtained by determining the open-loop Nash equilibria of an associated differential game with an additional initial state constraint. For the special case of linear-quadratic differential games, we derive both necessary and sufficient conditions for solvability of the finite planning horizon problem. In particular, we demonstrate that, unlike in the standard linear-quadratic differential game setting, uniqueness of equilibria may fail to hold. A both necessary and sufficient condition under which there is a unique equilibrium is provided. A sufficient existence condition for a unique equilibrium is derived in terms of a Riccati differential equation. Consequences for control policies are demonstrated in a simple debt stabilization game.

AB - This paper reconsiders existence of worst-case Nash equilibria in noncooperative multi-player differential games, this, within an open-loop information structure. We show that these equilibria can be obtained by determining the open-loop Nash equilibria of an associated differential game with an additional initial state constraint. For the special case of linear-quadratic differential games, we derive both necessary and sufficient conditions for solvability of the finite planning horizon problem. In particular, we demonstrate that, unlike in the standard linear-quadratic differential game setting, uniqueness of equilibria may fail to hold. A both necessary and sufficient condition under which there is a unique equilibrium is provided. A sufficient existence condition for a unique equilibrium is derived in terms of a Riccati differential equation. Consequences for control policies are demonstrated in a simple debt stabilization game.

KW - robust control

KW - noncooperative differential games

KW - linear optimal control

KW - Riccati equations

U2 - 10.1002/oca.2290

DO - 10.1002/oca.2290

M3 - Conference contribution

SN - 9789036763219

SP - 200

EP - 207

BT - Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014)

PB - University of Groningen

CY - Groningen

ER -

Engwerda JC. Robust open-loop Nash equilibria in the non-cooperative LQ game revisited. In Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014). Groningen: University of Groningen. 2014. p. 200-207 https://doi.org/10.1002/oca.2290