# Game Theory-Based Smart Mobile-Data Offloading Scheme in 5G Cellular Networks
## 4 Problem Formulation for MDO Based on NBS with Game Model  
The MDO problem based on game theory with an NBS is defined by \(\mathcal{G}=\left(\mathcal{I},S_{i},\phi_{i}\right),\forall i\in\mathcal{I}, \mathcal{I}=\left\{1,2,\ldots,I\right\}\) where \(I\) is the number of players and is also the number of link pairs between UE\({}_{m}\) and UE\({}_{n}\). \(S^{*}\) is a set of possible strategies for each players, and \(\Phi=\left(\phi_{1},\ldots,\phi_{i},\ldots,\phi_{I}\right)\) is a set of the payoffs for link pairs between UE\({}_{m}\) and UE\({}_{n}\), where \(\phi_{i}\) is the payoff function for player \(p_{i}\). The payoff for each player represents the cost that player \(p_{i}\) must endure for taking an action, \(S^{opt}_{i}=\omega^{opt}_{mn}\). \(i\). **Players**\((p_{i})\): The link pairs between UE\({}_{m}\) and UE\({}_{n}\), \(\mathcal{P}=\left(p_{1},\ldots,p_{i},\ldots,p_{I}\right)\), \(\forall i\in\mathcal{I}\). **Strategies**\(\left(S^{*}\right)\): Each link pair of \(m\) and \(n\) has a set of possible actions, \(S^{*}=\left(S_{1}^{opt},\ldots,S_{i}^{opt},\ldots S_{I}^{opt}\right)\), where \(S_{i}^{opt}=\omega_{mn}^{opt},\forall m,n\in\mathcal{M},n\
eq m\), and \(0\leq\omega_{mn}^{opt}\leq 1\) that represents the strategy space for player \(p_{i}\). **Payoff function**\(\phi_{i}\left(\lambda_{mn}\right)\): This defines the total cost for a link pair between UE\({}_{m}\) and UE\({}_{n}\) at amount of data of \(\lambda_{mn}\) in a mobile environment. The payoff function is defined to include the profit (the utility function), the energy consumption, and a network connecting cost (i.e., monetary cost) presented as follows:  
* **Monetary cost**\(\left(C_{mn}\right)\) represents the cost the UE pays based on the maximum \(K\) amount of data the UE uses on any given provider. Because the payoff function is calculated based on different functions that include the utility function, energy consumption function, and monetary cost, these functions should be transferred into the normalized form. In normalized form, it is assumed that if a UE uses the amount of data \(K\) for transmission, the monetary cost should be transferred to a payoff unit. Therefore, in this paper, if a UE uses the amount of data \(\lambda_{mn}\) for transmission, the monetary cost transferred to a payoff unit as follow: \[C_{mn}=\frac{\lambda_{mn}}{K}\] (15)
* **Utility function**\(\left(U_{mn}\left(\lambda_{mn}\right)\right)\) represents the profit of player \(p_{i}\) for using strategy \(\omega_{mn}\): \[U_{mn}\left(\lambda_{mn}\right)=\alpha\log\left(\lambda_{mn}+C\right)-\log \left(\lambda_{req}^{min}\right)\] (16) where \(\lambda_{mn}\) is the total amount of data of UE\({}_{m}\), which is given in (6); \(\alpha\) is a user-defined factor; \(C\) is a safety constant to make sure there is always a defined value for the utility function; and \(\lambda_{req}^{min}\) is the required minimal transmission data. In normalized form, if UE\({}_{m}\) uses amount of data \(\lambda_{mn}\) for transmission to UE\({}_{n}\), the utility function will be transferred to a payoff unit (\(C_{U_{mn}}\)), which is used to calculate the payoff of each link pair. The cost of utility function is represented as follows: \[C_{U_{mn}}=\frac{U_{mn}\left(\lambda_{mn}\right)}{U_{mn}\left(K\right)}\] (17) where \(U_{mn}\left(K\right)\) is the normalized function with amount of data \(K\) for the utility function which is defined by \(U_{mn}\left(K\right)=\log\left(K+C\right)\). * **Energy consumption function**\(\left(E_{s}\left(\lambda_{mn}\right)\right)\) is one of the most important factors in many network applications with a high cost to replace batteries. When data packets are sent from the source node to the sink node, energy consumption is generated. More packet transmissions means a higher data rate and higher energy consumption. Another factor that affects energy consumption is the density of the network; the more UEs with additional packet transmissions, the higher the density of the network, and thus, the higher the energy consumption.