# Marginal Contribution-Based Distributed Subchannel Allocation in Small Cell Networks
## 1 Introduction  
Small cells, which are typically characterized as providing dynamic range coverage and low-powered radio access nodes overlaid on macrocells, provide more flexible and scalable deployment by reusing available radio resources [1]. Deployment of small cells decreases the load of macrocell base stations (MBSs), provides a better link with a reduced distance between serving base stations (BSs) and mobile stations (MSs), and enhances network performance with improved capacity. However, the advantages of small cells could be insufficient in dense and random networks due to strong co-channel interference from neighboring small cells utilizing common radio resources, such as subchannels in orthogonal frequency division multiple access (OFDMA) networks [2]. Resource allocation becomes a critical issue and a challenge as it induces interference in the network, restricting both capacity and cellular coverage [3]. Centralized approaches for resource allocation [4; 5; 6; 7] require some central authority to maintain complete knowledge of the environment through frequent information exchange with small cell base stations (SBSs). However, such requirements may not be desirable when considering randomly deployed large numbers of small cells, due to the computational complexity of the centralized solution [8]. As the number and location of small cells become unpredictable, mobile operatorsshould consider distributed approaches allowing SBSs to self-organize by learning their dynamic environment, and gradually allocate resources. Game theory [9], as a distributed approach for resource allocation, is suitable for modeling the individualistic behaviors of small cells, where interactions among multiple decision-makers (players) over common resources can be formulated and analyzed as a game. Recently, researchers have proposed distributed learning rules for the problem of resource allocation [10; 11; 12; 13; 14; 15; 16], which can be generalized in two ways: (1) as uncoordinated; and (2) as coordinated. Some uncoordinated learning rules, such as spatial adaptive play [10], utility-based learning [11], stochastic-learning-automata [12], and reinforcement learning [13], are completely uncoupled, i.e., the resource allocation procedure does not depend on the actions of anyone else. With restriction to information exchange, the achieved solutions of these approaches have a trade-off in terms of convergence speed and efficiency. Coordinated learning rules, for example, cloud-assisted best-response [14] and cluster formation [15; 16], require some coordination mechanism such that a player is selected to opportunistically update its strategy at a given time. Although it maintains fairness by providing equal opportunity to the decision-makers, such a greedy adaption-based resource allocation, in general, does not necessarily result in a steady-state solution. Nevertheless, the distributed approaches have the advantages of scalable implementation, but their performance is typically inferior compared to the centralized approaches. In this regard, while distributed decision-makers are opportunistically trying to maximize their own performance, it is necessary to implement some mechanisms in their learning rules such that stable resource allocation can be obtained, while efficiency and fairness are also maintained within the network. In this paper, we focus on coordinated learning rules for resource allocation and model OFDMA-based small cell networks (SCNs) consisting of SBSs as distributed decision-makers trying to select a proper subchannel for transmission. To guarantee for existence of a steady-state solution, we resort to a class of potential game (PG) called the exact potential game (EPG) [17], and utilize the concept of marginal contribution (MC) [8] as a mechanism to design the welfare-based learning rule to achieve efficient and fair resource allocations.