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conference paper

Nash equilibria in scalar discrete-time linear quadratic games

Salizzoni, Giulio  
•
Ouhamma, Reda
•
Kamgarpour, Maryam  
June 24, 2025
2025 European Control Conference (ECC)
2025 European Control Conference (ECC)

An open problem in linear quadratic (LQ) games has been characterizing the Nash equilibria. This problem has renewed relevance given the surge of work on understanding the convergence of learning algorithms in dynamic games. This paper investigates scalar discrete-time infinite-horizon LQ games with two agents. Even in this arguably simple setting, there are no results for finding all Nash equilibria. By analyzing the best response map, we formulate a polynomial system of equations characterizing the linear feedback Nash equilibria. This enables us to bring in tools from algebraic geometry, particularly the Gröbner basis, to study the roots of this polynomial system. Consequently, we can not only compute all Nash equilibria numerically, but we can also characterize their number with explicit conditions. For instance, we prove that the LQ games under consideration admit at most three Nash equilibria. We further provide sufficient conditions for the existence of at most two Nash equilibria and sufficient conditions for the uniqueness of the Nash equilibrium. Our numerical experiments demonstrate the tightness of our bounds and showcase the increased complexity in settings with more than two agents.

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Type
conference paper
DOI
10.23919/ecc65951.2025.11187226
Author(s)
Salizzoni, Giulio  

École Polytechnique Fédérale de Lausanne

Ouhamma, Reda

École Polytechnique Fédérale de Lausanne

Kamgarpour, Maryam  

École Polytechnique Fédérale de Lausanne

Date Issued

2025-06-24

Publisher

IEEE

Published in
2025 European Control Conference (ECC)
DOI of the book
https://doi.org/10.23919/ECC65951.2025
Start page

2416

End page

2421

Editorial or Peer reviewed

REVIEWED

Written at

EPFL

EPFL units
SYCAMORE  
Event nameEvent acronymEvent placeEvent date
2025 European Control Conference (ECC)

ECC 2025

Thessaloniki, Greece

2025-06-24 - 2025-06-27

FunderFunding(s)Grant NumberGrant URL

National Science Foundation

Available on Infoscience
October 16, 2025
Use this identifier to reference this record
https://infoscience.epfl.ch/handle/20.500.14299/254984
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