Nash Equilibrium Seeking in Quadratic Noncooperative Games Under Two Delayed Information-Sharing Schemes
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Nash Equilibrium Seeking in Quadratic Noncooperative Games Under Two Delayed Information-Sharing Schemes Tiago Roux Oliveira1 · Victor Hugo Pereira Rodrigues2 Miroslav Krsti´c3 · Tamer Ba¸sar4
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Received: 31 March 2020 / Accepted: 22 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we propose non-model-based strategies for locally stable convergence to Nash equilibrium in quadratic noncooperative games where acquisition of information (of two different types) incurs delays. Two sets of results are introduced: (a) one, which we call cooperative scenario, where each player employs the knowledge of the functional form of his payoff and knowledge of other players’ actions, but with delays; and (b) the second one, which we term the noncooperative scenario, where the players have access only to their own payoff values, again with delay. Both approaches are based on the extremum seeking perspective, which has previously been reported for real-time optimization problems by exploring sinusoidal excitation signals to estimate the Gradient (first derivative) and Hessian (second derivative) of unknown quadratic functions. In order to compensate distinct delays in the inputs of the players, we have employed predictor feedback. We apply a small-gain analysis 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 for the unknown quadratic payoffs 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. Keywords Extremum seeking · Nash equilibrium · (Non)cooperative games · Time delays · Predictor feedback · Averaging in infinite dimensions Mathematics Subject Classification 91A10 · 34K33 · 35Q93 · 93D05 · 93C35 · 93C40 Abbreviation ES Extremum seeking ODE Ordinary differential equation PDE Partial differential equation FDE Functional differential equation ISS Input-to-state stability
Extended author information available on the last page of the article
123
Journal of Optimization Theory and Applications
1 Introduction Game theory provides a theoretical framework for conceiving social situations among competing players and using mathematical models of strategic interaction among rational decision-makers [1,2]. Game-theoretic approaches to designing, modeling, and optimizing emerging engineering systems, biological behaviors and mathematical finance make this topic of research an extremely important tool in many fields with a wide range of applications [3–5]. Indeed, we can find numerous results for both theory and practice in the corresponding literature of differential games [6–12]. In this context, we can view games roughly in two categories: cooperative and noncooperative games [13]. A game is cooperative if the players are able to form binding commitments externally enforced (e.g., through contract law), resulting in collective p
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