Fictitious Play and Best-Response Dynamics in Identical Interest and Zero-Sum Stochastic Games

This paper proposes an extension of a popular decentralized discrete-time learning procedure when repeating a static game called fictitious play (FP) (Brown, 1951; Robinson, 1951) to a dynamic model called discounted stochastic game (Shapley, 1953). Our family of discrete-time FP procedures is proven to converge to the set of stationary Nash equilibria in identical interest discounted stochastic games. This extends similar convergence results for static games (Monderer & Shapley, 1996a). We then analyze the continuous-time counterpart of our FP procedures, which include as a particular case the best-response dynamic introduced and studied by Leslie et al. (2020) in the context of zero-sum stochastic games. We prove the converge of this dynamics to stationary Nash equilibria in identical-interest and zero-sum discounted stochastic games. Thanks to stochastic approximations, we can infer from the continuous-time convergence some discrete time results such as the convergence to stationary equilibria in zero-sum and team stochastic games (Holler, 2020).