Nash equilibrium seeking for dynamic systems with non-quadratic payoffs

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.