Nash Equilibrium Seeking in Quadratic Noncooperative Games Under Two Delayed Information-Sharing Schemes

Tiago Roux Oliveira (), Victor Hugo Pereira Rodrigues (), Miroslav Krstić () and Tamer Başar ()
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Tiago Roux Oliveira: State University of Rio de Janeiro (UERJ)
Victor Hugo Pereira Rodrigues: Federal University of Rio de Janeiro (UFRJ/COPPE)
Miroslav Krstić: University of California at San Diego (UCSD)
Tamer Başar: University of Illinois at Urbana-Champaign

Journal of Optimization Theory and Applications, 2021, vol. 191, issue 2, No 15, 700-735

Abstract: 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; 91A10; 34K33; 35Q93; 93D05; 93C35; 93C40 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10957-020-01757-z

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