## 3 Subsidy and Pricing Strategy of Electric Vehicle Sharing Based on Game Theory
### Payoff to Customers  
Assuming that consumers are rational individuals, according to the utility theory in economics, they are certain to choose the most effective travel mode. This section builds utility functions for shared and private travel. Assume that the users mainly consider three factors: Travel expenses, travel time and comfort [21]. Calculating the time cost and comfort cost, the three factors are converted into the same dimension for comparison to determine the strategy selected by the user [22]. The main target group for sharing electric vehicles is those who have a car purchase plan or a private car with low idle rate. By comparing the travel costs of shared and private car travel, consumers can make decisions intuitively, thus delaying the purchase plan or even selling the existing vehicles. It will significantly reduce vehicle ownership, speed the urban traffic and alleviate infrastructure shortage. The coefficient of comfort cost (COM) is calculated as follows [23]:  
\[COM=\frac{\omega\theta\alpha\times\sum_{j=1}^{6}d_{j}\big{(}\tilde{t}/60\big{)} /24}{\tilde{l}}, \tag{1}\]  
where \(\omega\) is the coefficient of time value; \(\theta\) is the time value of the area; \(\alpha\) is the reduction factor of the vehicle for restoring fatigue time; \(d_{j}\) is the score corresponding to the level of the j-th comfort influencing factor. The selected six indicators are ride stability, the perception of crowding in vehicle, privacy, air and temperature in vehicle, physical exertion and punctuality of the ride. Where \(\tilde{t}\) is the average travel time of the area, \(\tilde{l}\) is the average travel distance of the area.  
* Costs for electric vehicle sharing \[U_{sn}=(p_{s}s_{n}+p_{t}t_{n})+(t_{wn}+t_{n})p_{ttw}+COM_{s}s_{n},\] (2)  
Figure 2: Framework of the two-stage Stackelberg game model.  
where \(s_{n}\) and \(t_{n}\) are the travel miles and time of the travel n respectively; \(COM_{e}\) is the coefficient of comfort cost for electric vehicle sharing; \(t_{wn}\) is the waiting time including the time to borrowing point and from the returning point to the destination; \(p_{trn}\) is the time value of the user. Assume that the latter two satisfy uniform distribution [24].  
\[\mathrm{f}(p_{trn})=\frac{1}{\theta_{2}-\theta_{1}}, \tag{3}\]  
\[\mathrm{f}(t_{tun})=\frac{1}{\theta_{4}-\theta_{3}}, \tag{4}\]  
where \(\theta_{2}\) and \(\theta_{1}\) are the upper and lower limits of the user's time value respectively; \(\theta_{4}\) and \(\theta_{3}\) are the upper and lower limits of the waiting time respectively.  
\(\bullet\) Costs for private car  
\[U_{pn}=p_{p}s_{n}+t_{np}t_{tun}+COM_{p}s_{n}, \tag{5}\]  
where \(p_{p}\) is the coefficient of private car travel expenses, including fuel consumption, maintenance, tire wear, insurance, depreciation, etc; \(COM_{p}\) is the coefficient of comfort cost for private cars.  
\(\bullet\) Payoff of the consumers considering the travel choice  
\[U_{C}=-x_{n}^{1}U_{sn}-x_{n}^{2}U_{pn}, \tag{6}\]  
\[\mathrm{s.t.}\ x_{n}^{1}+x_{n}^{2}=1, \tag{7}\]  
\[x_{n}^{1},x_{n}^{2}\in\{0,1\}, \tag{8}\]  
where \(x_{n}\) is the decision strategy; \(x_{n}^{1}=1\) means electric vehicle sharing is chosen during travel n; and \(x_{n}^{2}=1\) means private car is chosen.