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Learning in stochastic games is a notoriously difficult problem because, inaddition to each other's strategic decisions, the players must also contendwith the fact that the game itself evolves over time, possibly in a verycomplicated manner. Because of this, the convergence properties of popularlearning algorithms - like policy gradient and its variants - are poorlyunderstood, except in specific classes of games (such as potential ortwo-player, zero-sum games). In view of this, we examine the long-run behaviorof policy gradient methods with respect to Nash equilibrium policies that aresecond-order stationary (SOS) in a sense similar to the type of sufficiencyconditions used in optimization. Our first result is that SOS policies arelocally attracting with high probability, and we show that policy gradienttrajectories with gradient estimates provided by the REINFORCE algorithmachieve an $\mathcal{O}(1/\sqrt{n})$ distance-squared convergence rate if themethod's step-size is chosen appropriately. Subsequently, specializing to theclass of deterministic Nash policies, we show that this rate can be improveddramatically and, in fact, policy gradient methods converge within a finitenumber of iterations in that case.

On the convergence of policy gradient methods to Nash equilibria in general stochastic games