For tolerant containment control of multi-agent systems, considering the challenges in modeling and the impact of actuator faults on system security and reliability, a finite index dynamic event-triggered policy iteration algorithm is proposed. This algorithm only requires input and output data, without relying on system models, and simultaneously considers the faults and energy consumption issues to improve the system reliability and save energy consumption. The conditions are provided to demonstrate the convergence and optimality of the algorithm, including a convergence speed, that is, the number of iterations required for convergence is finite. For the convenience of practical implementation, an actor-critic structure is adopted and an actor network weight tuning law with actuator fault factors is designed to more accurately approximate the control protocol in policy iteration algorithm. In addition, an event-triggered mechanism is employed in saving computational resources. Finally, simulation results verify the efficiency of the designed algorithm.