Nash Equilibrium Seeking with Arbitrarily Delayed Player Actions

Tiago Roux Oliveira, Victor Hugo Pereira Rodrigues, Miroslav Krstic, Tamer Basar

Research output: Chapter in Book/Report/Conference proceedingConference contribution


In this paper, we propose a non-model based strategy for locally stable convergence to Nash equilibrium in a quadratic noncooperative (duopoly) game with arbitrarily delayed player actions. In our noncooperative scenario, the players have access only to their own payoff values, again with delay. The proposed approach is based on the extremum seeking perspective, which has previously been reported for real-time optimization problems by exploring sinusoidal perturbation signals to estimate the Gradient (first derivative) and Hessian (second derivative) of unknown locally quadratic functions. Indeed, this is the first contribution which considers extremum seeking for noncooperative games in the presence of delays. In order to compensate distinct delays in the inputs of the two players, we have employed boundary control via predictor feedback with averaging-based estimates. We apply a small-gain analysis for the resulting Input-to-State Stable hyperbolic PDEODE loop as well as averaging theory in infinite dimensions, due to the infinite-dimensional state of the time delays, in order to obtain local convergence results to a small neighborhood of the Nash equilibrium. We quantify the size of these residual sets and corroborate the theoretical results numerically on an example of a two-player game with delays.

Original languageEnglish (US)
Title of host publication2020 59th IEEE Conference on Decision and Control, CDC 2020
PublisherInstitute of Electrical and Electronics Engineers Inc.
Number of pages6
ISBN (Electronic)9781728174471
StatePublished - Dec 14 2020
Event59th IEEE Conference on Decision and Control, CDC 2020 - Virtual, Jeju Island, Korea, Republic of
Duration: Dec 14 2020Dec 18 2020

Publication series

NameProceedings of the IEEE Conference on Decision and Control
ISSN (Print)0743-1546
ISSN (Electronic)2576-2370


Conference59th IEEE Conference on Decision and Control, CDC 2020
Country/TerritoryKorea, Republic of
CityVirtual, Jeju Island

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization


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