Learning in concave games with imperfect information

This paper examines the convergence properties of a class of learning schemes for concave N -person games – that is, games with convex action spaces and individually concave payoff functions. Specifically, we focus on a family of learning methods where players adjust their actions by taking small steps along their individual payoff gradients and then “mirror” the output back to their feasible action spaces. Assuming players only have access to gradient information that is accurate up to a zero-mean error with bounded variance, we show that when the process converges, its limit is a Nash equilibrium. We also introduce an equilibrium stability notion which we call variational stability (VS), and we show that stable equilibria are locally attracting with high probability whereas globally stable states are globally attracting with probability 1. Additionally, in finite games, we find that dominated strategies become extinct, strict equilibria are locally attracting with high probability, and the long-term average of the process converges to equilibrium in 2-player zero-sum games. Finally, we examine the scheme’s convergence speed and we show that if the game admits a strict equilibrium and the players’ mirror maps are surjective, then, with high probability, the process converges to equilibrium in a finite number of steps, no matter the level of uncertainty.