Device-to-device (D2D) communication is a promising technology to improve energy efficiency and spectrum efficiency of the next-generation mobile networks. In this paper, we propose a D2D data off-loading scheme using game theoretical approach to reduce energy consumption for a wireless mobile network service provider. In the meantime, our proposed scheme provides a fair incentive mechanism to motivate D2D relay users for participation. The D2D scenario studied in this work focuses on downlink communications from the service provider to some users who request the same service, e.g., live streaming of a sports game. As an incentive, the service provider rewards some data usage to D2D relay users. In particular, we formulate a Stackelberg game to find the optimal portion of off-loading data for the relay users and the optimal incentive mechanism settings for the base station. With the proposed D2D off-loading scheme, a base station can maximize the energy saving while providing the most attraction to D2D relay users. The simulation results demonstrate that our proposed scheme will enhance the network energy efficiency and be attractive to D2D relay users.