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Conventional Generative Adversarial Networks (GANs) for text generation tend to have issues of reward sparsity and mode collapse that affect the quality and diversity of generated samples. To address the issues, we propose a novel self-adversarial learning (SAL) paradigm for improving GANs' performance in text generation. In contrast to standard GANs that use a binary classifier as its discriminator to predict whether a sample is real or generated, SAL employs a comparative discriminator which is a pairwise classifier for comparing the text quality between a pair of samples. During training, SAL rewards the generator when its currently generated sentence is found to be better than its previously generated samples. This self-improvement reward mechanism allows the model to receive credits more easily and avoid collapsing towards the limited number of real samples, which not only helps alleviate the reward sparsity issue but also reduces the risk of mode collapse. Experiments on text generation benchmark datasets show that our proposed approach substantially improves both the quality and the diversity, and yields more stable performance compared to the previous GANs for text generation.