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We analyze the convergence properties of the two-timescale fictitious playcombining the classical fictitious play with the Q-learning for two-playerzero-sum stochastic games with player-dependent learning rates. We show itsalmost sure convergence under the standard assumptions in two-timescalestochastic approximation methods when the discount factor is less than theproduct of the ratios of player-dependent step sizes. To this end, we formulatea novel Lyapunov function formulation and present a one-sided asynchronousconvergence result.

On the Heterogeneity of Independent Learning Dynamics in Zero-sum Stochastic Games