We examine so-called product-games. These are n-player stochastic games played on a product state space s 1 × .. × s n , in which player i controls the transitions on s i . For the general n-player case, we establish the existence of 0-equilibria. In addition, for the case of two-player zero-sum games of this type, we show that both players have stationary 0-optimal strategies. In the analysis of product-games, interestingly, a central role is played by the periodic features of the transition structure. Flesch et al. (math oper res 33, 403–420, 2008) showed the existence of 0-equilibria under the assumption that, for every player i, the transition structure on s i is aperiodic. In this article, we examine product-games with periodic transition structures. Even though a large part of the approach in flesch et al. (math oper res 33, 403–420, 2008) remains applicable, we encounter a number of tricky problems that we have to address. We provide illustrative examples to clarify the essence of the difference between the aperiodic and periodic cases.