  We analyze the convergence properties of the two-timescale fictitious play
combining the classical fictitious play with the Q-learning for two-player
zero-sum stochastic games with player-dependent learning rates. We show its
almost sure convergence under the standard assumptions in two-timescale
stochastic approximation methods when the discount factor is less than the
product of the ratios of player-dependent step sizes. To this end, we formulate
a novel Lyapunov function formulation and present a one-sided asynchronous
convergence result.
