Before revision:  In the pure strategies case, we prove that every two player finite action game has at least one periodic strategy, making the periodic strategies an inherent characteristic of these games .
After revision:  In the pure strategies case, we prove that every two player finite action game has at least one periodic strategy, with non-trivial meaning a game with non-degenerate payoffs. In principle the algorithm we provide, holds true for every non-trivial game, because in degenerate games, inconsistencies can occur. In addition, we also address the incomplete information games in the context of Bayesian games, in which case generalizations of Bernheim's rationalizability offers us the possibility to embed the periodicity concept in the Bayesian games framework .