Existing methods for image captioning are usually trained by cross entropy loss, which leads to exposure bias and the inconsistency between the optimizing function and evaluation metrics. Recently it has been shown that these two issues can be addressed by incorporating techniques from reinforcement learning, where one of the popular techniques is the advantage actor-critic algorithm that calculates per-token advantage by estimating state value with a parametrized estimator at the cost of introducing estimation bias. In this paper, we estimate state value without using a parametrized value estimator. With the properties of image captioning, namely, the deterministic state transition function and the sparse reward, state value is equivalent to its preceding state-action value, and we reformulate advantage function by simply replacing the former with the latter. Moreover, the reformulated advantage is extended to n-step, which can generally increase the absolute value of the mean of reformulated advantage while lowering variance. Then two kinds of rollout are adopted to estimate state-action value, which we call self-critical n-step training. Empirically we find that our method can obtain better performance compared to the state-of-the-art methods that use the sequence level advantage and parametrized estimator respectively on the widely used MSCOCO benchmark.