## 4 Proposed Control and Incentive Mechanism
### Mechanism of the Incentives
Similarly, the social externality of EV driver connected to the \(i\)th terminal of the charging station (the optimal social program if agent \(i\) is absent) can be defined as [31; 37; 38]:  
\[W_{t}^{-i}(\mathbf{b}_{t}):=\inf_{\mathbf{a}_{-i,t}}\sum_{j\
eq i}c_{jt}\big{(}\mathbf{b}_ {i\prime},\mathbf{a}_{jt}\big{)}+c_{0t}(\mathbf{a}_{-i,t},\gamma), \tag{24}\]  
with the socially efficient policy \(\mathbf{a}_{-i,t}^{*}\) if \(i\) is absent. Then the change in the social value due to the addition of agent \(i\) can be defined by its marginal contribution at time \(t\)[31; 37; 38]:  
\[m_{i,t}(\mathbf{b}_{t})=W_{t}(\mathbf{b}_{t})-W_{t}^{-i} \tag{25}\]  
So, the payment rule, such that the total discomfort experienced by the \(i\)th agent (22) equals their marginal contribution, can be defined as follows [31; 37; 38]:  
\[p_{i,t}^{*}(\mathbf{b}_{t},\mathbf{a}_{t}^{*})\triangleq c_{it}(\mathbf{a}_{it}^{*},\mathbf{b }_{it})-m_{i,t}(\mathbf{b}_{t}) \tag{26}\]  
By substituting (21) and (25) into the (26), we can express the payment policy for EV drivers connected to the \(i\)th terminal of the charging station in terms of social welfare functions of agents as follows [31; 37; 38]:  
\[p_{i,t}^{*}=\sum_{j\
eq i}c_{j,t}\big{(}\mathbf{a}_{-i,t}^{*},\mathbf{b}_{jt}\big{)}+c _{0t}\big{(}\mathbf{a}_{-i,t}^{*},\gamma_{t}\big{)}-\sum_{j\
eq i}c_{j,t}\big{(} \mathbf{a}_{t}^{*},\mathbf{b}_{jt}\big{)}-c_{0t}(\mathbf{a}_{t}^{*},\gamma_{t}). \tag{27}\]  
**Proposition 1**. : _The payment policy defined in (27) is incentive compatible and individually rational._  
**Proof of Proposition 1. ** By the unprovability principle, it suffices to prove that if agent \(i\) receives their marginal contribution as a payment, then truth-telling is incentive compatible for agent \(i\) in time \(t\), or that:  
\[l_{it}(\mathbf{\theta}_{it},\mathbf{a}_{it}^{*},\mathbf{\theta}_{t})\leq l_{it}(\mathbf{b}_{it },\mathbf{a}_{it}^{*},(\mathbf{b}_{it},\mathbf{\theta}_{-it})), \tag{28}\]  
where \(\mathbf{b}_{it}\
eq\mathbf{\theta}_{it}\). By the definition of \(l_{it}\) in (22) and \(p_{i,t}^{*}\) in (27), the left-hand side of (28) represents the marginal contribution of agent \(i\):  
\[m_{i,t}(\mathbf{\theta}_{t})\leq c_{it}\Big{(}\mathbf{\theta}_{it},\mathbf{a}_{it(\mathbf{b}_{ it},\mathbf{\theta}_{-it})}^{*}\Big{)}-p_{it}^{*}\Big{(}\big{(}\mathbf{b}_{it},\mathbf{ \theta}_{-it}),\mathbf{a}_{t(\mathbf{b}_{it},\mathbf{\theta}_{-it})}^{*}\Big{)} \tag{29}\]  
Then, the payment \(p_{i,t}^{*}\Big{(}(\mathbf{b}_{it},\mathbf{\theta}_{-it}),\mathbf{a}_{t(\mathbf{b}_{it},\mathbf{ \theta}_{-it})}^{*}\Big{)}\) can be inserted based on the definition in (27) into (29) and rearranged to obtain:  
\[m_{i,t}(\mathbf{\theta}_{t})\leq\sum_{i}c_{it}\Big{(}\mathbf{a}_{t(\mathbf{b}_{it},\mathbf{ \theta}_{-it})}^{*},\mathbf{\theta}_{it}\Big{)}-\sum_{j\
eq i}c_{jt}(\mathbf{a}_{-i,t}^ {*},\mathbf{\theta}_{it})+c_{0t}\Big{(}\mathbf{a}_{t(\mathbf{b}_{it},\mathbf{\theta}_{-it})}^{*},\gamma_{t}\Big{)}-c_{0t}(\mathbf{a}_{-i,t}^{*},\gamma_{t}) \tag{30}\]for all \(i\in\mathcal{I}\). Then, we can express (30) in terms of the social welfare (21), (24), and (25) as:  
\[W_{t}(\mathbf{\theta}_{t})-W_{t}^{-i}(\mathbf{\theta}_{t})\leq W_{t}((\mathbf{b}_{it},\mathbf{ \theta}_{-it}))-W_{t}^{-i}(\mathbf{\theta}_{t}) \tag{31}\]  
By adding \(W_{t}^{-i}(\mathbf{\theta}_{t})\) to both sides of the inequality (31), we get:  
\[L_{it}(\mathbf{\theta}_{it},\mathbf{a}_{it},\mathbf{b}_{-i,t})=\inf_{\mathbf{b}_{it}\in\mathbf{ \Theta}_{it}}l_{it}(\mathbf{\theta}_{it},\mathbf{a}_{it},\mathbf{b}_{t}) \tag{23}\]0  
The above inequality holds for all \(\mathbf{b}_{it}\) by the social optimality of \(\mathbf{a}_{t}^{*}\). \(\square\)  
**Proposition 2**. : _The defined mechanism \(\mathcal{M}(\mathbf{a}_{t}^{*}\,\ \mathbf{p}_{t}^{*})\) satisfies the efficient exit condition._  
**Proof of Proposition 2. ** By the efficient exit condition definition, the agents do not receive payments at the time steps in which they are irrelevant to the mechanism.