This paper deals with a class of dynamic discrete time open-loop games played over an infinite time horizon. The equilibrium concept is defined in the sense of overtaking optimal responses by the players to the program choices of the opponents. We extend to this dynamic game framework the results obtained by Rosen for concave static games. We prove existence, uniqueness, and asymptotic stability (also called the turnpike property) of overtaking equilibrium programs for a class of games satisfying a strong concavity assumption (strict diagonal concavity).