Log-linear learning: Convergence in discrete and continuous strategy potential games

In this paper, we consider log-linear learning algorithm in potential games. This algorithm can be applied to solving cooperative control problems in multi-agent systems. We investigate the convergence properties of the log-linear learning algorithm in potential games with discrete and continuous strategy sets. So far, the convergence of this algorithm 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 analyze the probabilistic convergence of the log-linear learning algorithm not only in discrete strategy games, but also in continuous strategy ones. We present a way of parameter setting that guarantees the probabilistic convergence of system behavior to the set of potential function maximizers in both cases. This result is valuable since in many settings the potential function maximizers correspond to the optimal states in the multi-agent system.