An integrated EPON/WiMAX network has been an attractive attention due to its features of achieving stable backhaul transmissions to WiMAX users and reducing the expenditure of network deployment. In implementing such a network, realizing a scheduling mechanism capable of sharing the available bandwidth between the heterogeneous Ethernet and WiMAX traffic in an effective manner is problematic. In the present study, this problem is addressed by using a game-theoretic-based scheme comprising a bankruptcy game and a bargaining game to perform an intra-ONU scheduling function at each ONU within the network. In the proposed approach, each ONU distributes the total amount of bandwidth allocated by the OLT between the Ethernet and WiMAX traffic buffers in accordance with the results of the bankruptcy game. By applying bargaining games, the allocated bandwidth is then further distributed to the different traffic classes within the Ethernet or WiMAX traffic in accordance with their QoS requirements. The simulation results confirm the validity and effectiveness of the proposed approach.