TY - GEN

T1 - Nash equilibrium seeking for dynamic systems with non-quadratic payoffs

AU - Frihauf, Paul

AU - Krstic, Miroslav

AU - Başar, Tamer

N1 - Funding Information:
This research was made with Government support under and awarded by DoD, Air Force Office of Scientific Research, National Defense Science and Engineering Graduate (NDSEG) Fellowship, 32 CFR 168a and by grants from the National Science Foundation, AFOSR and DOE.

PY - 2011

Y1 - 2011

N2 - We consider a general, stable nonlinear dynamic system with N inputs and N outputs, where in the steady state, the output signals represent the non-quadratic payoff functions of a noncooperative game played by the values of the input signals. We introduce a non-model based approach for locally stable convergence to a steady-state Nash equilibrium. In classical game theory algorithms, each player employs the knowledge of the functional form of its payoff and of the other players' actions, whereas in the proposed algorithm, the players need to measure only their own payoff values. This strategy is based on the so-called "extremum seeking" approach, which has previously been developed for standard optimization problems and employs sinusoidal perturbations to estimate the gradient. Since non-quadratic payoffs create the possibility of multiple, isolated Nash equilibria, our convergence results are local. Specifically, the attainment of any particular Nash equilibrium is assured only for initial conditions in a set around that specific stable Nash equilibrium. Moreover, for non-quadratic payoffs, the convergence to a Nash equilibrium is biased in proportion to the perturbation amplitudes and the payoff functions' third derivatives. We quantify the size of these residual biases.

AB - We consider a general, stable nonlinear dynamic system with N inputs and N outputs, where in the steady state, the output signals represent the non-quadratic payoff functions of a noncooperative game played by the values of the input signals. We introduce a non-model based approach for locally stable convergence to a steady-state Nash equilibrium. In classical game theory algorithms, each player employs the knowledge of the functional form of its payoff and of the other players' actions, whereas in the proposed algorithm, the players need to measure only their own payoff values. This strategy is based on the so-called "extremum seeking" approach, which has previously been developed for standard optimization problems and employs sinusoidal perturbations to estimate the gradient. Since non-quadratic payoffs create the possibility of multiple, isolated Nash equilibria, our convergence results are local. Specifically, the attainment of any particular Nash equilibrium is assured only for initial conditions in a set around that specific stable Nash equilibrium. Moreover, for non-quadratic payoffs, the convergence to a Nash equilibrium is biased in proportion to the perturbation amplitudes and the payoff functions' third derivatives. We quantify the size of these residual biases.

KW - Extremum seeking

KW - Nash equilibria

KW - Noncooperative games

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U2 - 10.3182/20110828-6-IT-1002.00232

DO - 10.3182/20110828-6-IT-1002.00232

M3 - Conference contribution

AN - SCOPUS:84866753274

SN - 9783902661937

T3 - IFAC Proceedings Volumes (IFAC-PapersOnline)

SP - 3605

EP - 3610

BT - Proceedings of the 18th IFAC World Congress

PB - IFAC Secretariat

ER -