# A Hierarchical Resource Allocation Scheme Based on Nash Bargaining Game in VANET
## 2 Hierarchical Resource Allocation Scheme Based on Nash Bargaining Game
### Bargaining Game and Nash Bargaining Solution  
The game theory can effectively solve the problem of competition among the VUEs in the process of resource allocation. Although the selfish characteristics of VUEs in non-cooperative games conform to the actual situation, the rational behaviors of VUEs will damage the overall benefit of the system and restrict the benefit of other VUEs. Therefore, relevant constraints need to be established among the VUEs to enable them to cooperate with each other and to improve the overall efficiency of the system as much as possible. Bargaining game [16] is a kind of cooperative game in which both players have the opportunity to reach a win-win situation. In this game, there is a conflict of interest among the participants. If either party cannot accept the bargaining scheme, the overall agreement will not be reached. Before the specific definition is given, a simple example of two-person bargaining game is given firstly. We assume participant 1 and participant 2 share the benefits of the system, which is \(X\). Participant 1 first proposes a share of the proceeds, which is \(x\), \(x\in[0,X]\). In addition, then, participant 2 judges whether to accept the agreement based on the proposal and decision of participant 1. If the agreement is accepted, participant 2 receives the remaining proceeds, which is \(X-x\). Otherwise, participant 2 rejects the agreement, both parties fail to reach an agreement, the benefits will be zero. The definition of a Nash bargaining game is as follows. \(G=(K,\mathcal{S},\mathcal{U},\mathbf{u}^{0})\) represents a game problem with \(K\) players. \(\mathcal{S}\) represents the set of all participants' policies. The set of benefits that participants can obtain in the game is \(\mathcal{U}\). \(u_{0}^{j}\in\mathbf{u}^{0}=\{u_{1}^{0},u_{2}^{0},...,u_{K}^{0}\}\) represents the benefit of the \(i\)thparticipant under the failure agreement. Clearly, participants do not participate in the collaboration when the benefits allocated to the collaboration are less than \(u_{i}^{0}\). Then, game \(G\) is called the bargaining game of \(K\) participants. The Nash Bargaining Solution (NBS) is the function mapping that can make the above bargaining game \(G\) has the unique optimal return vector, \(\mathbf{u}^{*}=\{u_{1}^{*},u_{2}^{*},...,u_{K}^{*}\}=f(\mathcal{U},\mathbf{u} ^{0})\). However, there may be multiple bargaining functions, therefore, the most widely used bargaining function is selected to realize effective and reasonable allocation of resources in VANET. This bargaining solution is subject to the following conditions [17]. 1. \(\mathbf{u}^{*}\in\mathcal{U}\). 2. \(\mathbf{u}^{*}\) is Pareto optimal. 3. As for any linear mapping \(g\), it should satisfy \(f(g(\mathcal{U}),g(\mathbf{u}^{0}))=g(f(\mathcal{U},\mathbf{u}^{0}))\). 4. If \(\mathbf{v}\subset\mathcal{U}\) and \(f(\mathcal{U},\mathbf{u}^{0})\in\mathcal{U},f(\mathcal{U},\mathbf{u}^{0})=f( \mathbf{v},\mathbf{u}^{0})\). 5. Symmetry. Conditions 1 and 2 guarantee the validity and rationality of the existence of NBS, condition 2 means the ideal state of resource allocation, in which the bargaining scheme is optimal. Conditions 3 to conditions 5 guarantee the proportional fairness of NBS, and conditions 3 indicates that NBS has zoom invariance. It means that if the benefit function changes linearly, the final NBS stays the same. The condition 4 indicates that the expansion of the domain will not affect the final result of the bargaining game. The bargaining function of NBS meeting the above five conditions is shown below  
\[f(\mathcal{U},\mathbf{u}^{0})\in\max_{u_{i}\in\mathcal{U},u_{i}>u^{0}}\prod _{i=1}^{K}(u_{i}-u^{0}) \tag{1}\]  
\(\prod_{i=1}^{K}(u_{i}-u^{0})\) is a strictly quasiconcave function on a nonempty closed set \(\mathcal{U}\). This function has a unique maximum value in the maximum solution problem.