In this paper, the near-Nash equilibrium (NE) control strategies are investigated for a class of discrete-time nonlinear systems subjected to the round-robin protocol (RRP). In the studied systems, three types of complexities, namely, the additive nonlinearities, the RRP, and the output feedback form of controllers, are simultaneously taken into consideration. To tackle these complexities, an approximate dynamic programing (ADP) algorithm is first developed for NE control strategies by solving the coupled Bellman's equation. Then, a Luenberger-type observer is designed under the RRP scheduling to estimate the system states. The near-NE control strategies are implemented via the actor-critic neural networks. More importantly, the stability analysis of the closed-loop system is conducted to guarantee that the studied system with the proposed control strategies is bounded stable. Finally, simulation results are provided to demonstrate the validity of the proposed method.