Fast convergence in semi-anonymous potential games

The log-linear learning algorithm has been extensively studied in game theoretic and distributed control literature. A central appeal of log-linear learning for distributed control is that it often guarantees agents´ behavior will converge in probability to the optimal configuration. However, one of its central issues is that the worst case convergence time can be prohibitively long, e.g., exponential in the number of players. We formalize a modified log-linear learning algorithm whose worst case convergence time is roughly linear in the number of players. We prove this characterization in semi-anonymous potential games with limited populations, i.e., in potential games where the agents´ utility functions can be expressed as a function of aggregate behavior within each population.