Nash equilibrium seeking for games with non-quadratic payoffs

We introduce a non-model based approach for asymptotic, locally stable attainment of Nash equilibria in static noncooperative games with N players. In classical game theory algorithms, each player employs the knowledge of both the functional form of its payoff and the other players' actions. The proposed algorithm, in which the players only measure their own payoff values, 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. We consider static games where the players seek to maximize their non-quadratic payoff functions. 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 not assured for all initial conditions, but only for initial conditions in a set around that specific stable Nash equilibrium. For non-quadratic payoffs, the convergence to a Nash equilibrium is not perfect, but is biased in proportion to the perturbation amplitudes and the third derivatives of the payoff functions. We quantify the size of these residual biases.