We examine product-games, which are n-player stochastic games satisfying: (1) the state space is a product S(1)×…×S(n); (2) the action space of any player i only depends of the i-th coordinate of the state; (3) the transition probability of moving from s(i) ∈ S(i) to t(i) ∈S(i), on the i-th coordinate S(i) of the state space, only depends on the action chosen by player i. So, as far as the actions and the transitions are concerned, every player i can play on the i-th coordinate of the product-game without interference of the other players. No condition is imposed on the payoff structure of the game. We focus on product-games with an aperiodic transition structure, for which we present an approach based on so-called communicating states. For the general n-player case, we establish the existence of 0-equilibria, which makes product-games one of the first classes within n-player stochastic games with such a result. In addition, for the special case of two-player zero-sum games of this type, we show that both players have stationary 0-optimal strategies. Both proofs are constructive by nature.
