Proving convergence of log-linear learning in potential games

In this paper, we provide a theoretical analysis of log-linear learning algorithm that can be used for studying decision processes and solving control problems related to potential games. So far, for this algorithm the convergence of collective behavior to some state in a potential game has been proven to be specified by a chosen parameter and does not imply the convergence in probability to potential function maximizers. Tending the parameter and time to infinity simultaneously, we investigate the probabilistic convergence of joint actions based on the log-linear learning algorithm. We formulate conditions under which the convergence does not take place at all. Nevertheless, we explain how the parameter and utility functions should be designed to guarantee the probabilistic convergence of system behavior with some stationary distribution. Moreover, for such a setup in potential games the stable states, i.e. those having positive probability in the limit stationary distribution, are from the set of Nash equilibria and maximize the potential function.