## 4 Taxonomy: Classification of GT Approaches in Network Virtualization
### Game Goals
#### 4.2.1 Goal 1: Maximize the Bandwidth Allocation  
The virtualization of the resources such as the bandwidth introduces critical challenges to be addressed and solved in the NV environment. Many solutions as static allocation are proposed to provide a sufficient bandwidth assignment by satisfying all users' demands. The static allocation technique aims to provide fixed bandwidth to each virtual link by assigning the demanded bandwidth when VN is created. Due to the network dynamically, the static bandwidth allocation isn't efficient anymore and suffers from many limitations and problems by leading to the degradation of the network performance. Dynamically sharing the bandwidth is a suitable solution to overcome the static allocation issues, but it remains an open problem due to its instability. According to the game theory's dynamicity and flexibility, applying this mathematical tool for bandwidth allocation can be an up-and-coming solution. In [17], the authors applied a particular type of game theory called the Stackelberg game to allocate dynamically the bandwidths between the virtual networks competing for the physical resources. During the Stackelberg game, the leader first selects the strategy, and then the follower chooses the appropriate action by reacting to the leader's decision. Based on the Stackelberg game principle, the authors considered the virtual networks as a follow by playing a non-cooperative game to maximize their gain in bandwidth allocation, and the substrate network as leader by setting a price to drive virtual networks to maximize revenue. A Stackelberg equilibrium is proposed to reach the optimal solution for each player by minimizing the price of allocation. This equilibrium point offers an efficient and fair bandwidth allocation. The authors in [18] seek to treat the bandwidth allocation problem for the network virtualization. As a solution, they applied the GT to model the interaction between the InPs and the SPs deploying the customized Virtual Network (VN). Thus, they are the first contribution that presented the first paper which developed a particular case for applying the GT in the network virtualization environment. In this competitive environment, the InP aims to maximize its revenue by allocating SPs' physical resources, seeking to allocate sufficient resources. The complex interaction between the InP and the SPs can be modeled as a non-cooperative and finite game to uniformly and efficiently allocate the physical resources among the VNs owned by the SPs. They introduced a robust payoff function based on three sub-functions: utility function, pricing function, and congestion function defined by the following formula:  
\[\Phi_{k}=U_{k}(y^{k},z^{k})-P_{k}(y^{k})-C_{k}(y^{k},z^{k}) \tag{2}\]  
The utility function is defined by \(U_{k}\) which presents the gain obtained by the k-th VN from allocating the bandwidth \(y_{k}\) with a predefined path rate denoted as \(z_{k}\). The pricing function \(P_{k}\) computes the total cost of assigned bandwidths for the corresponding VNs. Finally, the congestion function \(C_{k}\) measures the cost of congestion for the assigned bandwidths and current path rate. If the path rate of all VNs is less than the capacity of the link, the congestion cannot therefore appear. On the other hand, if the leasing of the bandwidth by the VN leads to the congestion of the link, the concerning VN must pay the cost of the occurred congestion. The next step, in the proposed scheme, is to prove the existing of the Nash equilibrium. One of the solutions is to prove the convexity of the payoff function [55]. Therefore, the authors find that the convexity is related to the congestion cost which is a convex function. Through Nash equilibrium convergence, all the players can achieve the best solution: \"Maximize the utility function and optimize the cost function\". An iterative algorithm is proposed to reach the Nash equilibrium state. The convergence to Nash equilibrium is proved by a simple topology composed of two nodes and two links.