## 1 Introduction  
The standard solution for supporting multicast services in Long Term Evolution-Advanced (LTE-A) networks suffers from severe inefficiencies [1], as the conventional cellular multicast performance is strictly bounded by the user with the poorest channel conditions. For this reason, researchers are active in studying solutions to enhance the multicast services performance in cellular systems [1]. Among many alternatives, another new paradigm within LTE-A systems, namely Device-to-Device (D2D) communications [2], has become a flourishing research field. D2D communication is about enabling direct flow of data among User Equipments (UEs) in close proximity, it holds merit of data offloading for the eNodeB (eNB), improving spectrum efficiency, and increasing system capacity [2; 3; 4], _etc._ These observations led us to the idea of bringing D2D communications into the cellular multicast services framework, and investigating the potentialities of adopting D2D communications to enhance the performance of conventional multicast services. In LTE-A networks, UEs requesting the same data from eNB can be grouped into a cluster and eNB multicasts data to all UEs within the cluster. The major issue in cellular multicast transmissions is the different channel conditions for the multicast recipients. Alternatively in D2D-assisted networks, UEs with good channel conditions are enabled to act as relays to communicate the received multicast data from the eNB to the remaining UEs via D2D links, such that the eNB can multicast at a high data rate and the transmission efficiency is improved. Based on this idea, in [5] the authors proposed a cooperative multicast transmission scheme, _i.e._, in which all successful multicast recipients participate in D2D relay transmissions by broadcasting their received data on the same resource. Because each D2D relay should serve all failed multicast recipients at a common rate selected according to the worst D2D link, it is unfavorable to improve the network throughput. In [6] the authors assumed a single predefined multicast recipient called cluster head which is responsible for D2D relay transmissions. Unfortunately the efficiency of this scheme is not guaranteed in case the predesignated recipient fails to receive multicast data and the eNB must participate in retransmissions. In [7] the authors proposed an adaptive D2D retransmission scheme exploiting the multi-channel diversity to improve spectrum, which adaptively selects the optimal number of D2D retransmitters and conducts the optimal sub-cluster partition. Furthermore, in [8] authors introduced relay based transmission into the multicast process so as to overcome transmission rate restriction caused by link heterogeneity, and a greedy heuristic algorithm was applied to select relay nodes. According to the these works, to improve the transmission quality of intra-cluster multicast service, a high efficient relay selection should be considered in the D2D cluster. In this paper we consider an auction-based approach using game theory. A combinatorial auction (CA) model for resource management was introduced in [9]. Combinatorial auctions are multi-item or multi-bidder auctions in which bidders can form combinations called packages, rather than just bid individually. Furthermore, in the evolution auction mechanisms named iterative combinatorial auctions [10], the step of bid evaluation is executed multiple times, and the auctioneer computes provisional allocations in each auction round. Iterative CAs have already found applications in allocating radio spectrum for wireless communications [11]. The CAs motivate bidders to fully express their preferences, which is an advantage in improving system efficiency and auction revenues. Up to that point, we study an effective D2D relay selection for multicast services to further improve system efficiency based on the iterative CA. The whole system consists of the eNB, multiple successful multicast recipients, and multiple failed multicast recipients.