Despite the success of generative adversarial networks (GANs) in generating visually appealing images, they are notoriously challenging to train. In order to stabilize the learning dynamics in minimax games, we propose a novel recursive reasoning algorithm: Level $k$ Gradient Play (Lv.$k$ GP) algorithm. Our algorithm does not require sophisticated heuristics or second-order information, as do existing algorithms based on predictive updates. We show that as k increases, Lv.$k$ GP converges asymptotically towards an accurate estimation of players' future strategy.Moreover, we justify that Lv.$\infty$ GP naturally generalizes a line of provably convergent game dynamics which rely on predictive updates. Furthermore, we provide its local convergence property in nonconvex-nonconcave zero-sum games and global convergence in bilinear and quadratic games. By combining Lv.$k$ GP with Adam optimizer, our algorithm shows a clear advantage in terms of performance and computational overhead compared to other methods. Using a single Nvidia RTX3090 GPU and 30 times fewer parameters than BigGAN on CIFAR-10, we achieve an FID of 10.17 for unconditional image generation within 30 hours, allowing GAN training on common computational resources to reach state-of-the-art performance.