In this paper, we newly model computation offloading competition when multiple clients compete with each other so as to reduce energy cost and improve computational performance. We consider two types of destination of offloading request such as a cloudlet and a remote cloud. Here, the cloudlet consists of locally connected mobile terminals with low-latency and high bandwidth but suffering from task overload due to its limited computational capacity. On the other hand, the remote cloud has a high and stable capacity but the high latency. To facilitate the competition model, on the destination sides, we have designed an energy-oriented task scheduling scheme which aims to maximize the welfare of clients in terms of energy efficiency. Under this proposed job scheduling, as a joint consideration of the destination and client sides, competition behavior among multiple clients for optimal computation offloading is modeled and analyzed as a non-cooperative game by considering a trade-off between different types of destinations. Based on this game-theoretical analysis, we propose a novel energy-oriented weight assignment scheme in the mobile terminal side to maximize mobile terminal energy efficiency. Finally, we show that the proposed scheme converges well to a unique equilibrium and it maximizes the payoff of all participating clients.