Flexible Nash seeking using stochastic difference inclusions

We present a novel algorithm designed to achieve robust convergence to Nash equilibria in non-cooperative games, where players are not required to participate in the game for all time, neither to know the exact mathematical form of their cost function. In this algorithm each player employs stochastic probing dynamics that only require measurements of its own cost function, together with a dynamic time-ratio mechanism that enforces its frequency of participation in the game to satisfy a time-ratio constraint. The algorithm is modeled by a constrained stochastic difference inclusion with non-unique solutions that encompass a complete set of admissible behaviors for each player. To characterize the convergence and stability properties of the system we introduce the notion of mean-square practical exponential stability for constrained stochastic difference inclusions, as well as sufficient Lyapunov conditions that certify this property. Simulation examples are used to demonstrate the results.