Dynamic Adaptive Streaming over HTTP (DASH) has demonstrated to be an emerging and promising multimedia streaming technique, owing to its capability of dealing with the variability of networks. Rate adaptation mechanism, a challenging and open issue, plays an important role in DASH based systems since it affects Quality of Experience (QoE) of users, network utilization, etc. In this paper, based on non-cooperative game theory, we propose a novel algorithm to optimally allocate the limited export bandwidth of the server to multi-users to maximize their QoE with fairness guaranteed. The proposed algorithm is proxy-free. Specifically, a novel user QoE model is derived by taking a variety of factors into account, like the received video quality, the reference buffer length, and user accumulated buffer lengths, etc. Then, the bandwidth competing problem is formulated as a non-cooperation game with the existence of Nash Equilibrium that is theoretically proven. Finally, a distributed iterative algorithm with stability analysis is proposed to find the Nash Equilibrium. Compared with state-of-the-art methods, extensive experimental results in terms of both simulated and realistic networking scenarios demonstrate that the proposed algorithm can produce higher QoE, and the actual buffer lengths of all users keep nearly optimal states, i.e., moving around the reference buffer all the time. Besides, the proposed algorithm produces no playback interruption.