# Capacity Analysis of Incentive Schemes in Opportunistic Networks
## 2 Mathematical Model for Incentive Schemes
### Quantum Game Model  
The proposed model has two parts, as shown in Figure 1. The quantum game model depicts how an incentive scheme motivates cooperation, and the utility formula \(\mathfrak{S}_{m}\) describes how a node is served. The utility formula \(\mathfrak{S}_{m}\), including the energy usage, buffer usage, and degree of security, accepts an incentive policy IP and entanglement \(\lambda\) as arguments. In both the quantum game model and the utility formula, \(\lambda\) communicates with a quantum game model to describe incentive processes. Therefore, the optimal value \(\mathfrak{S}^{*}\) is computed by the quantum game model with Nash equilibrium.  
Quantum game theory is more suitable for analyzing incentive schemes because of the concept of entanglement and the extended strategy space, as discussed in Section 1. First, we briefly describe quantum game theory based on [11], shown in Figure 2. The actions of players in a quantum game ae based on a specific set of strategies. The state of node \(m\) is represented by \(\ket{vac_{m}}\), a state vector in a Hilbert space. The whole system state is \(\ket{H}=\ket{H_{1}}\otimes\ket{H_{2}}\cdots\otimes\ket{H_{n}}\).  
Figure 1: Model construction.  
Figure 2: Quantum game model.  
If players are dependent, entanglement must be used to express the system state in the above product. Entangling gate \(\hat{\mathbf{f}}(\lambda)\) with operation parameter \(x_{m}\) defines how to combine the strategies of each node. This expresses how players affect others, such as encounters due to moving paths and selfish policies, especially in opportunistic networks; in our study, entanglement is considered as the cooperation degree. The model includes the following three stages:  
1. The initial quantum state of the system is \(\psi_{init}=\hat{\mathbf{f}}(\lambda)|vac\rangle\), achieved by entangling all nodes togather.
2. Node \(m\) operates on its state using unitary operator \(\hat{D}_{m}\) with the feature \(\hat{D}_{m}\hat{D}_{m}^{\dagger}=1\).
3. With disentangling gate \(\hat{J}(\lambda)^{\dagger}\) on the system, the final state is \(\left|\psi_{fin}\right\rangle=|J\rangle(\lambda)^{\dagger}(\hat{D_{1}}\otimes \hat{D_{2}}\cdots\hat{D_{n}})\hat{J}(\lambda)|vac\rangle\).  
Each definition and proof are detailed in [11]. After forwarding \(\left|\psi_{fin}\right\rangle\) to measurement, node \(m\) can quantize the relayed data to others as follows:  
\[q_{m}=\frac{x_{m}}{n}\Big{[}e^{(n-1)\lambda}+(n-1)e^{-\lambda}\Big{]}+\sum_{ j=1,j\
eq m}^{n}\frac{x_{j}}{n}\Big{[}e^{(n-1)\lambda}-e^{-\lambda}\Big{]}\,. \tag{1}\]  
The optimal value \(x_{m}^{*}\) can be computed by a utility formula with the Nash equilibrium for impacts on nodes. The Nash equilibrium is defined as follows:  
\[\begin{cases}\frac{\partial\hat{\mathbf{5}}_{1}}{\partial x_{1}}=\frac{\partial \hat{\mathbf{5}}_{2}}{\partial x_{2}}=\cdots=\frac{\partial\hat{\mathbf{5}}_{n}}{ \partial x_{n}}=0\\ \frac{\partial^{2}\hat{\mathbf{5}}_{m}}{\partial x_{m}^{2}}<0\,,\,m\in\{1,2,\cdots,n\}\,.\end{cases} \tag{2}\]  
Then, the optimal strategy \(q_{m}^{*}\) and optimal utility \(\hat{\mathbf{5}}_{m}^{*}\) can be calculated with \(x_{m}^{*}\) according to (1) and (3). The calculation depends on the utility formula, with the input incentive scheme to be discussed later. Then, we obtain a mathematical equation between an incentive scheme and \(\hat{\mathbf{5}}^{*}\), which can be used for evaluatation and reveal the insight if the utility formula depicts factors correctly. Moreover, \(\hat{\mathbf{5}}^{*}\) is called the incentive degree in this study.