In this section, we use the backward induction method of game theory to analyze the two games and find the NE and the SE.
5.1 Subgame nash equilibrium
We use the backward induction method to analyze the existence and uniqueness of the NE in the subgame.
Theorem 2
There exists a NE point in the noncooperative subgame among vehicles.
Proof
The strategy space of each vehicle is nonempty, convex, and compact. From Eq. (7), \(U_i\) is continuous with respect to \(\epsilon\) in \([\epsilon _{\min },\epsilon _{\max }]\). We take the first and second derivatives of \(U_i\) with respect to \(\epsilon _i\) and obtain
$$\begin{aligned} &\frac{\partial U_i}{\partial \epsilon _i} = \frac{\frac{R}{v_i} {\sum _{j \in N \backslash \{ i \}}} \frac{\epsilon _j}{v_j} }{\left( \sum _{i \in N}\frac{\epsilon _i}{v_i} \right) ^2}  \left( c_i+\frac{e}{v_i} \right) , \\&\frac{\partial ^2 U_i}{\partial \epsilon _i^2} = \frac{\frac{2R}{v_i^2} {\sum _{j \in N \backslash \{ i \}}} \frac{\epsilon _j}{v_j}}{\left( \sum _{i \in N}\frac{\epsilon _i}{v_i} \right) ^3} < 0. \end{aligned}$$
(14)
We prove that \(U_i\) is strictly concave with respect to \(\epsilon _i\). Thus, the NE point exists. The proof is now completed. \(\square\)
Let \(\frac{\partial U_i}{\partial \epsilon _i}=0\) and we get the best response function of i as
$$\begin{aligned} \begin{aligned} \epsilon _i^* = {\left\{ \begin{array}{ll} \epsilon _{\min }, &{} R< {\underline{R}} \\ \sqrt{\frac{Ra_iv_i}{k_i}}  a_i v_i, &{} {\underline{R}} \le R < {\overline{R}} \\ \epsilon _{\max }, &{} R \ge {\overline{R}}, \end{array}\right. }, \end{aligned} \end{aligned}$$
(15)
where \(a_i = {\sum _{j \in N \backslash \{ i \}}} \frac{\epsilon _j}{v_j}\), \(k_i = c_i+\frac{e}{v_i}\), \({\underline{R}} = \frac{k_iv_i\left( a_i+\frac{\epsilon _{\min }}{v_i} \right) ^2}{a_i}\), and \({\overline{R}} = \frac{k_iv_i\left( a_i+\frac{\epsilon _{\max }}{v_i} \right) ^2}{a_i}\).
Theorem 3
At the NE point for the noncooperative subgame among the vehicles, the best response of i has a closedfrom expression given by
$$\begin{aligned} \epsilon _i^* = \frac{R v_i(N1)}{\sum _{i \in N} v_i k_i} \left( 1  \frac{v_i k_i(N1)}{\sum _{i \in N} v_i k_i} \right) . \end{aligned}$$
(16)
Proof
According to Eq. (15), we have
$$\begin{aligned} {\sum _{j \in N \backslash \{ i \}}} \frac{\epsilon _j}{v_j} = \frac{v_i k_i}{R} \left( \sum _{i \in N}\frac{\epsilon _i}{v_i} \right) ^2. \end{aligned}$$
(17)
By computing the summation of this expression for all the vehicles, we obtain
$$\begin{aligned} \sum _{i \in N}\frac{\epsilon _i}{v_i} = \frac{R(N1)}{\sum _{i \in N}v_i k_i}. \end{aligned}$$
(18)
We substitute Eq. (18) into Eq. (17) and get
$$\begin{aligned} \frac{R(N1)}{\sum _{i \in N}v_i k_i}  \frac{\epsilon _i}{v_i} = \frac{v_i k_i}{R} \left( \frac{R(N1)}{\sum _{i \in N}v_i k_i} \right) ^2. \end{aligned}$$
(19)
which can be rewritten as Eq. (16). The proof is now completed. \(\square\)
Theorem 4
The NE for the noncooperative subgame is unique if the following condition is satisfied.
$$\begin{aligned} \sum _{i \in N} v_i k_i > 2 v_i k_i (N1). \end{aligned}$$
(20)
Proof
According to Eqs. (17) and (18), we have
$$\begin{aligned} {\sum _{j \in N \backslash \{ i \}}} \frac{\epsilon _j}{v_j} = v_i k_i R \left( \frac{(N1)}{\sum _{i \in N}v_i k_i} \right) ^2. \end{aligned}$$
(21)
Given R offered by the SP and privacy strategies \(\pmb {\epsilon _{i}}\) offered by other vehicles, the best response function in Eq. (15) is denoted as \(\epsilon _i^* = B_i(\pmb {\epsilon _{i}},R)\). The NE is unique if \(B(\pmb {\epsilon },R) = (B_1,B_2,\dots ,B_N)\) can be proved to be the standard function which meets the following conditions [5, 29].

Positivity: \(B(\pmb {\epsilon },R) > 0\),

Monotonicity: For all \(\pmb {\epsilon }\) and \(\pmb {\epsilon }^{\prime}\), \(B(\pmb {\epsilon },R) \ge B(\pmb {\epsilon }^{\prime},R)\) if \(\pmb {\epsilon } \ge \pmb {\epsilon }^{\prime}\),

Scalability: For all \(\mu > 1\), \(\mu B(\pmb {\epsilon },R) > B(\mu \pmb {\epsilon },R)\).
We first analyze the positivity. According to Eq. (20), we have \(\frac{N1}{\sum _{i \in N} v_i k_i} < \frac{1}{2 v_i k_i}\), and thus conclude that
$$\begin{aligned} {\sum _{j \in N \backslash \{ i \}}} \frac{\epsilon _j}{v_j}< \frac{R}{4 v_i k_i} < \frac{R}{v_i k_i}. \end{aligned}$$
(22)
We further conclude that \({\sum _{j \in N \backslash \{ i \}}} \frac{\epsilon _j}{v_j} < \sqrt{\frac{R}{v_i k_i} {\sum _{j \in N \backslash \{ i \}}} \frac{\epsilon _j}{v_j} }\). Thus, we have
$$\begin{aligned} B_i(\pmb {\epsilon _{i}},R) = v_i \left( \sqrt{\frac{R}{v_i k_i}{\sum _{j \in N \backslash \{ i \}}} \frac{\epsilon _j}{v_j}}  {\sum _{j \in N \backslash \{ i \}}} \frac{\epsilon _j}{v_j} \right) > 0, \end{aligned}$$
(23)
which satisfies the positivity condition.
We then analyze the monotonicity. Taking the first derivative of \(B_i(\epsilon _{i},R)\) with respect to \(\epsilon _j\),\(j \in N \backslash \{ i \}\), we have
$$\begin{aligned} \frac{\partial B_i(\pmb {\epsilon _{i}},R)}{\partial \epsilon _j} = \frac{v_i}{v_j} \left( \frac{1}{2} \sqrt{\frac{R}{v_i k_i} \frac{1}{{\sum _{j \in N \backslash \{ i \}}} \frac{\epsilon _j}{v_j}} } 1 \right) . \end{aligned}$$
(24)
According to Eq. (22) that \({\sum _{j \in N \backslash \{ i \}}} \frac{\epsilon _j}{v_j} < \frac{R}{4 v_i k_i}\), we have \(\frac{1}{2} \sqrt{\frac{R}{v_i t_i} \frac{1}{{\sum _{j \in N \backslash \{ i \}}} \frac{\epsilon _j}{v_j}} } 1 > 0\). Thus, the monotonicity condition is satisfied.
Finally we analyze the scalability. We have
$$\begin{aligned} \mu B_i(\pmb {\epsilon _{i}},R)  B_i(\mu \pmb {\epsilon _{i}},R) = v_i(\mu  \sqrt{\mu }) \sqrt{\frac{R}{v_i k_i}{\sum _{j \in N \backslash \{ i \}}} \frac{\epsilon _j}{v_j}} \ge 0. \end{aligned}$$
(25)
Therefore, \(\mu B_i(\pmb {\epsilon _{i}},R) \ge B_i(\mu \pmb {\epsilon _{i}},R)\) is always satisfied for \(\mu > 1\). The scalability condition is satisfied. \(B(\epsilon _{i},R)\) meets the three conditions and is a standard function. Thus, uniqueness of the NE is proved. The proof is now completed.
\(\square\)
Generally, we can obtain the NE point by using the best response dynamics [29] . Problem 1 is resolved and then we analyze the SE in the following.
5.2 Stackelberg equilibrium
We substitute \(\epsilon _i^*\) into the objective function of Problem 2 and have
$$\begin{aligned} U_S = \lambda a \sum _{i \in N} \frac{1}{v_i} \log (h_i R + 1)  R, \end{aligned}$$
(26)
where \(h_i = \frac{b v_i(N1)}{\sum _{i \in N} v_i k_i} \left( 1  \frac{v_i k_i(N1)}{\sum _{i \in N} v_i k_i} \right)\).
Theorem 5
There exists a unique SE for the proposed Stackelberg game among the SP and the vehicles.
Proof
The strategy space of the SP is nonempty, convex, and compact. \(U_S\) is continuous with respect to R in \([0,+\infty ]\). We take the second derivatives of Eq. (26) with respect to R and get
$$\begin{aligned} \frac{\partial ^2 U_S}{\partial R^2} = \lambda a \sum _{i \in N} \frac{ h_i^2}{v_i(h_i R + 1)^2} < 0 . \end{aligned}$$
(27)
Thus, \(U_S\) is strictly concave with respect to R and the SP has a unique optimal strategy \(R^*\) in maximizing its utility. According to Theorem 4, given any reward from the SP, the vehicles always choose a unique set of best responses \(\epsilon ^*\) to reach the NE. Therefore, when the SP chooses \(R^*\), all players determine their optimal strategies. This satisfies the condition in Definition 2 that there exists a unique SE point. The proof is now completed. \(\square\)
The objective function of Problem 2 is a concave function and can be solved by using the existing typical convex optimal algorithms (e.g., dual decomposition algorithm [30] ). If the SP has global information, such as \(c_i\), he can find out \(R^*\) in a centralized manner. However, to protect the privacy of each vehicle, [31] inspires us to design a distributed algorithm that performs the optimization without any private information. The proposed incentive mechanism is carried out cyclically. At each cycle, the SP and the vehicles reach an agreement by Algorithm 3. Under the agreement, the vehicles finish the coinference tasks by choosing a privacy budget and obtaining the responding rewards. In Algorithm 3, the SP updates the reward value by using a gradientassisted searching algorithm, i.e., Eq. (28), and offers it to the vehicles. Each vehicle receives the reward value, determines its privacy budget based on Eq. (15), and returns the strategy to the SP. The iterations continue until the difference of the updated reward value is less than a preset threshold. Note that the communication delay is negligible due to the small size of shared information. The frequency of update, i.e., the number of iterations to reach convergence, depends on the learning rate and the threshold. When executing the algorithm, the vehicles conduct wireless communication with an access point (AP). Each vehicular node uploads its strategy information, i.e., privacy budget \(\epsilon _i\), to the nearest AP and other vehicular nodes can query this strategy information with negligible delay.