Fictitious play in zero-sum stochastic games

We present fictitious play dynamics for the general class of stochastic games and analyze its convergence properties in zero-sum stochastic games. Our dynamics involves agents forming beliefs on opponent strategy and their own continuation payoff (Q-function), and playing a myopic best response using estimated continuation payoffs. Agents update their beliefs at states visited from observations of opponent actions. A key property of the learning dynamics is that update of the beliefs on Q-functions occurs at a slower timescale than update of the beliefs on strategies. We show both in the model-based and model-free cases (without knowledge of agent payoff functions and state transition probabilities), the beliefs on strategies converge to a stationary mixed Nash equilibrium of the zero-sum stochastic game.