## 4 Incentive Mechanism for Contracts of Vehicles
### Contract Feasibility  
For feasibility, each contract must satisfy the following conditions:  
**Definition 1**. : _Individual rationality (IR): Each VU only participates in the federated learning task when the utility of the VU is not less than zero, i.e.,_  
\[\varepsilon_{j}R_{j}-cq_{j}\geq 0 \tag{4}\]  
**Definition 2**. : _Incentive compatibility (IC): Each VU of type-j only chooses the contract designed for its type, i.e., \(\left(R_{j},q_{j}\right)\) instead of any other contracts \(\left(R_{z},q_{z}\right)\) to maximize utility, i.e.,_  
\[\varepsilon_{j}R_{j}-cq_{j}\geq\varepsilon_{j}R_{z}-cq_{z} \tag{5}\]  
**Lemma 1**. : _Monotonicity: For contract \(\left(R_{j},q_{j}\right)\) and \(\left(R_{z},q_{z}\right)\), we can see that \(R_{j}\geq R_{z}\), if and only if \(\varepsilon_{j}\geq\varepsilon_{z}\) and \(j\
eq z\), \(j,z\in\left\{1,\ldots,J\right\}\)._Proof. : Based on the IC constraints of type-\(m\) VU and type-\(z\) VU, we can see that  
\[\varepsilon_{j}R_{j}-cq_{j}\geq\varepsilon_{j}R_{z}-cq_{z} \tag{6}\]  
\[\varepsilon_{z}R_{z}-cq_{z}\geq\varepsilon_{z}R_{j}-cq_{j} \tag{7}\]  
This paper first proves adequacy: adding Equations (6) and (7) by transformation, we can obtain  
\[\varepsilon_{j}R_{j}+\varepsilon_{z}R_{z}\geq\varepsilon_{j}R_{z}+\varepsilon_ {z}R_{j} \tag{8}\]  
\[\varepsilon_{j}R_{j}-\varepsilon_{z}R_{j}\geq\varepsilon_{j}R_{z}-\varepsilon_ {z}R_{z} \tag{9}\]  
Combining Equations (8) and (9), we can obtain \(R_{j}\big{(}\varepsilon_{j}-\varepsilon_{z}\big{)}\geq R_{z}\big{(} \varepsilon_{j}-\varepsilon_{z}\big{)}\). Note that \(\varepsilon_{j}-\varepsilon_{z}\geq 0\), thus it can be proven that \(R_{j}\geq R_{z}\). Similarly, this demonstrates necessity: we can obtain \(\varepsilon_{j}\big{(}R_{j}-R_{z}\big{)}\geq\varepsilon_{z}(R_{j}-R_{z})\). If \(R_{j}-R_{z}\geq 0\), it follows that \(\varepsilon_{j}\geq\varepsilon_{z}\). As such, Lemma 1 is proven. Lemma 2: _If the IR constraint of type-\(1\) is satisfied, then other IR constraints will also hold._  
_According to the IC constraint, \(\forall j\in\{2,...,J\}\), we can obtain_  
\[\varepsilon_{j}R_{j}-cq_{j}\geq\varepsilon_{j}R_{1}-cq_{1} \tag{10}\]  
_and, because \(\varepsilon_{1}<\cdots<\varepsilon_{j}<\cdots<\varepsilon_{J}\), we can obtain_  
\[\varepsilon_{j}R_{1}-cq_{1}\geq\varepsilon_{1}R_{1}-cq_{1} \tag{11}\]  
_Therefore, by combining Equations (10) and (11), we can obtain_  
\[\varepsilon_{j}R_{j}-cq_{j}\geq\varepsilon_{1}R_{1}-cq_{1} \tag{12}\]  
_Equation (12) indicates that when the IR constraint of type-\(1\) VUs is satisfied, other IR constraints will automatically remain unchanged. Therefore, other IR constraints can be limited to the IR conditions of type-\(1\) VUs._  
Lemma 3: _According to the monotonicity in Lemma 1, the IC constraints can be simplified to local downward incentive constraints (LDICs), expressed as follows:_  
\[\varepsilon_{j}R_{j}-cq_{j}\geq\varepsilon_{j}R_{j-1}-cq_{j-1} \tag{13}\]  
Proof. : The IC constraint between type-\(j\) and type-\(z\), \(\forall z\in\{1,\ldots,j-1\}\), is defined as a downward IC (DIC), expressed as \(\varepsilon_{j}R_{j}-cq_{j}\geq\varepsilon_{j}R_{z}-cq_{z}\). First, it is proven that the DIC can be reduced to two adjacent types of DIC, which are called LDICs.