Almost-Sure Convergence of a Class of Nonautonomous Fictitious Play

A nonautonomous version of continuous-time fictitious play is considered to achieve global asymptotic stability of a unique Nash equilibrium of a game. The proposed approach to prove global asymptotic stability consists of combining successively in time a continuous-time static fictitious play characterized by a time-varying rate of convergence with a continuous-time proportional derivative fictitious play, which has been recently proposed. Convergence to the basin of attraction of the empirical frequency of the proportional derivative fictitious play, if it exists, is obtained by means of contraction tools, reminiscent of the small-gain theorem, provided a set of inequalities involving the parameter of the continuous-time dynamics is satisfied. Furthermore, a discrete-time fictitious play is derived from its continuous-time counterpart. Convergence with probability one to the unique Nash equilibrium is shown. The approach is illustrated with a modified version of the Shapley game for which the proposed scheme is shown to be convergent to the unique equilibrium with the additional flexibility of selecting the rate of convergence