Due to the limited computing resource and battery capability at the mobile devices, the computation-intensive tasks generated by mobile devices can be offloaded to edge servers or cloud for processing. In this paper, we study the multi-user task offloading problem in an end-edge-cloud system, in which all user devices compete for the limited communication and computing resources. Particularly, we first formulate the offloading problem with the goal of maximizing the Quality of Experience (QoE) of the users subject to resource constraints. Since each user focuses on maximizing its own QoE, we reformulate the problem as a Multi-User Task Offloading Game (MUTO-Game). We then identify an important property that for any device, both the communication interference and the degree of computing resource competition can be upper bounded. Based on the property, we further theoretically prove that there exists at least one Nash Equilibrium offloading strategy in the MUTO-Game. We propose the Game-based Decentralized Task Offloading (GDTO) approach to obtain the Nash Equilibrium offloading strategy. Finally, we analyze the upper bound for the convergence time and characterize the performance guarantee of the obtained offloading strategy for the worst case. A series of experimental results are presented, in comparison with both the centralized optimal approach and the approximate approaches.