# Game Theory-Based Smart Mobile-Data Offloading Scheme in 5G Cellular Networks
## 4 Problem Formulation for MDO Based on NBS with Game Model
The energy consumption function is defined as follows: \[E_{s}\left(\lambda_{mn}\right)=\beta D_{me}\lambda_{mn}\] (18) where \(\beta\) is a user-defined factor given for the energy-saving requirement, and \(D_{me}\) is the density metric of the network. We can express the network density in terms of the number of UEs per nominal coverage area. Thus, if \(M\) UEs are scattered in area \(A\), and the nominal range of each UE is \(R\), the density metric will be given as follows [63]: \[D_{me}=\frac{\left|M\right|\pi R^{2}}{A}\] (19)where \(A\) is transmission area of macro BS which is defined by \(A=4r^{2}\) with \(r\) is transmission radius of macro BS. The cost of energy consumption function (\(C_{E_{s}}\)), which based on the normalized form, is calculated as:  
\[C_{E_{s}}=\frac{E_{s}\left(\lambda_{mm}\right)}{K} \tag{20}\]  
For each player \(p_{i}\), the payoff function based on the normalized form can be declared as:  
\[\phi_{i}\left(\lambda_{mm}\right)=C_{U_{mn}}-C_{E_{s}}-C_{mm} \tag{21}\]  
Let \(E_{re}\left(t\right)\) denote the remaining-energy function. The UE updates its remaining energy for time slot \(t+1\). \(E_{re}\left(t\right)\) is the amount of energy remaining in the battery in the \(t\)th time slot. When the updated energy of the UE is less than the energy consumption, the UE will not have enough energy to transmit data, and will harvest energy from an ambient non-RF signal. Conversely, if the UE has enough energy to transmit data (i.e., \(E_{re}\left(t\right)+e^{h\upsilon}\left(t\right)\geq E_{s}\)), it will transmit data to another UE. The updated remaining energy for the next time slot is calculated by:  
\[E_{re}\left(t+1\right)=\min\left\{\max\left\{E_{re}\left(t\right)+e^{h\upsilon }\left(t\right)-E_{s},0\right\},E_{but}\right\} \tag{22}\]  
When the remaining energy of the UE is updated, the payoff function of each player is also updated over \(t\) time slots. Then, in this paper, the final payoff is obtained by averaging the payoffs over \(N_{timeslot}\), which is used as a set of the payoffs with the NBS for the MDO problem. To find a solution to the game, \(\mathcal{G}=\left(\mathcal{I},S_{i},\phi_{i}\right),\forall i\in\mathcal{I}\), a proof that it has a unique solution is required, and this means that each player can reach an optimal strategy, \(S_{i}^{opt}=\omega_{mn}^{opt}\), where it has no incentive to change its strategy given that all other players maintain their current strategies. There exists a unique and optimal NBS, which was proved in [56]. The Nash bargaining problem, which determines optimal offload coefficient \(\omega_{mn}^{opt}\), such that the NBS payoff function can be maximized (for example, by using advanced novel optimization techniques proposed in [36; 64; 65]) for this game, is presented as follows:  
\[S^{*}=\max_{S_{i}^{opt}}\prod_{i=1}^{I}\left(\phi_{i}-\phi_{i}^{ min}\right)\] (23) s.t. \[\phi_{i}\geq\phi_{i}^{min} \tag{24}\] \[U_{mn}\left(\lambda_{mn}\right)=\alpha\log\left(\lambda_{mn}+C\right)-\log \left(\lambda_{req}^{min}\right)\]0  
Moreover, in order to evaluate how fairly the resources are distributed among existing D2D-link pairs, we use the [PERSON]'s fairness index [25] as a fairness index (FI) as follows:  
\[U_{mn}\left(\lambda_{mn}\right)=\alpha\log\left(\lambda_{mn}+C\right)-\log \left(\lambda_{req}^{min}\right)\]1  
In section of simulation results, we will evaluate the fairness of network with this fairness index. In addition, in the MFI scheme, one of baselines scheme, UEs will offload data traffic to another UE such that the fairness index of the network can be maximized as follows:  
\[U_{mn}\left(\lambda_{mn}\right)=\alpha\log\left(\lambda_{mn}+C\right)-\log \left(\lambda_{req}^{min}\right)\]2 (27) s.t. \[U_{mn}\left(\lambda_{mn}\right)=\alpha\log\left(\lambda_{mn}+C\right)-\log \left(\lambda_{req}^{min}\right)\]3