Logit-Q Dynamics for Efficient Learning in Stochastic Teams

We present a new family of logit-Q dynamics for efficient learning in stochastic games by combining the log-linear learning (also known as logit dynamics) for the repeated play of normal-form games with Q-learning for unknown Markov decision processes within the auxiliary stage-game framework. In this framework, we view stochastic games as agents repeatedly playing some stage game associated with the current state of the underlying game while the agents' Q-functions determine the payoffs of these stage games. We show that the logit-Q dynamics presented reach (near) efficient equilibrium in stochastic teams with unknown dynamics and quantify the approximation error. We also show the rationality of the logit-Q dynamics against agents following pure stationary strategies and the convergence of the dynamics in stochastic games where the stage-payoffs induce potential games, yet only a single agent controls the state transitions beyond stochastic teams. The key idea is to approximate the dynamics with a fictional scenario where the Q-function estimates are stationary over epochs whose lengths grow at a sufficiently slow rate. We then couple the dynamics in the main and fictional scenarios to show that these two scenarios become more and more similar across epochs due to the vanishing step size and growing epoch lengths.