## 1 Introduction  
The upcoming fifth-generation (5G) wireless communication system is expected to provide quality of service that fulfills the massive requirements of high data rates and the tremendous need for high connectivity and wireless data service. In such systems, excessive power consumption is inevitable and dealing with that dilemma is critical for mobile service operators. Currently, techniques such as orthogonal frequency division multiple access (OFDMA) have been exploited in the fourth generation (4G) wireless systems. In OFDMA, the frequency band is divided into orthogonal subchannels and each subchannel is assigned to one user at most [1]. As a promising technology, non-orthogonal multiple access (NOMA) is introduced for future wireless communication [2]. Unlike OFDMA, NOMA serves more than one user in a single subcarrier in a power domain. Consequently, the inter-user interference is inevitable, to deal with this problem; the successive interference cancelation (SIC) is employed at the receiver to decode the required information [3]. Since energy efficiency (EE) is a crucial measure of 5G, more studies have been investigating energy saving by designing efficient power allocation for different NOMA systems. In Reference [4], the authors introduced an energy efficient power allocation for a multi-carrier NOMA (MC-NOMA) system, with the proportional rate constraints to guarantee fairness among users. The authors of Reference [5] put fairness as a priority in designing an energy efficient power allocation scheme for NOMA in ultra-dense heterogeneous networks. The study in Reference [6] considered designing the power allocation algorithm under imperfect channel state information. In Reference [7], [PERSON] et al. proposed a joint resource allocation algorithm in hybrid MC-NOMA to achieve spectrum efficiency (SE)and EE, where an arbitrary number of users can be multiplexed on the same subcarrier. In Reference [8], subcarrier and power are allocated jointly, using a time-efficient allocation algorithm. The proposed resource allocation framework was designed to maximize the sum rate in three main steps. In the first step, the problem was relaxed by assuming all subcarriers were available to use for each user. In the second step, subcarriers were allocated to users and in the third step, a centralized power allocation was introduced to allocate power to the users. Energy efficient resource allocation in multi-carrier scenario was investigated in Reference [9], the authors initially used matching theory to allocate users to subchannels. Then, the EE maximization problem was divided into subproblems which was solved by using penalty function method. Finally, power and subchannels were jointly allocated. As an efficient and powerful method in strategic planning, game theory has been widely employed in resource allocation problems for wireless networks. In [PERSON] et al. [10], a non-cooperative super-modular game was designed for subchannel matching to improve energy efficiency, the power was allocated equally for each subchannel and the game utility function was designed with the aim of maximizing user's EE, based on that, subchannel matching was designed. The auction-mechanism is a part of game theory, which has played a significant role in resource allocation. The authors of Reference [11] first separated the resource of the 5G network and then they designed a hierarchical combinatorial auction mechanism framework for resource allocation. The authors of Reference [12], introduced an auction-based resource allocation for software-defined networks. The auction-based power allocation game between users in NOMA network was addressed by [PERSON] et al. [13]. The base station (BS) played the auctioneer while users are the bidders. Simulation results have shown the effectiveness and outperformed other benchmark algorithms. Table 1 illustrates the comparison of different related works. In this paper, we address the energy efficiency of MCMC-NOMA based networks with the constraints of rate requirements for users.