Regret minimization under partial monitoring

We consider repeated games in which the player, instead of observing the action chosen by the opponent in each game round, receives a feedback generated by the combined choice of the two players. We study Hannan consistent players for these games, that is, randomized playing strategies whose per-round regret vanishes with probability one as the number of game rounds goes to infinity. We prove a general lower bound for the convergence rate of the regret, and exhibit a specific strategy that attains this rate for any game for which a Hannan consistent player exists.