## 5 Nested Logit and Game Modeling Proposed
### Parking Pricing Analysis
#### 5.2.1. Reservation Parking Lot Pricing  
The costs of car parks with reserved parking are mainly divided into fixed costs (\(C_{b}\)) and variable costs (\(C_{v}\)). The fixed costs include the costs of operation, management, maintenance, and small rentals of the car parks, which are fixed costs using an average of the daily operations of the car parks and which do not vary with user demand or with the pricing of the remaining car parks. Variable costs refer to the fees paid by the garage for parking inspection, guidance information provided to users, etc., which vary with the volume of users entering the garage for parking and are variable values. Similarly, the variable cost is taken as the average cost per vehicle trip served per unit service cycle during the operating period of the garage (billed per vehicle trip per cycle). The cost of the garage for a single operating period is then:  
\[C_{x}=C_{b}+q_{x}C_{v} \tag{7}\]  
where \(C_{x}\) is the total cost of the xth operating period; \(C_{b}\) is the fixed cost of the garage; \(q_{x}\) is the total number of passengers entering the garage for parking in the x-th operating period (person trips); and \(C_{v}\) is the variable cost of the garage. Assuming that the maximum number of reservation periods/operating periods of the parking lot is n, the total cost of the operating period of the parking lot is:  
\[C=\sum\limits_{x=1}^{n}C_{x}=\sum\limits_{x=1}^{n}(C_{b}+q_{x}C_{v}) \tag{8}\]  
where \(C\) is the total cost of reserving the parking lot. The demand for a single operating period of the car park is:  
\[q_{x}=q\left(\sum\limits_{j=1}^{n_{x}}(P_{q_{jx}})\right)=q\left(\sum\limits_{ j=1}^{n_{x}}\frac{\exp(V_{a})}{\sum\limits_{k=1}^{3}\exp(V_{k})}.\frac{\exp(V_{jx})} {\sum\
olimits_{l=1}^{n!}\exp(V_{l})}\right) \tag{9}\]  
where \(q_{x}\) is the total parking demand (person-times) in the x-th operation period of the car park; \(P_{axj}\) is the probability that the user chooses the combination of reservation periods \(j_{x}\) (including the x-th service period); a total of \(n_{x}\) combinations include the x-th operation period. The total parking demand in the surrounding mall area during the operating hours of the car park \(Q\) (person trips) according to the above equation enables us to calculate the total demand of the reserved parking lot, as:  
\[q=Q\cdot p_{a}=Q\frac{\exp(V_{a})}{\sum\limits_{k=1}^{3}\exp(V_{k})} \tag{10}\]  
where \(p_{a}\) is the probability that the user chooses to reserve parking. The total profit during the operation of the reserved parking lot is:  
\[ \begin{array}{c}Z=\sum\limits_{x=1}^{n}(w_{x}q_{x}-C_{x})=\sum\limits_{x=1} ^{n}(q_{x}(w_{x}-C_{v})-C_{b})\\ =\sum\limits_{x=1}^{n}\left(Q\frac{\exp(V_{a})}{\sum\limits_{k=1}^{3}\exp(V_{k })}\left(\sum\limits_{j=1}^{n_{x}}\frac{\exp(V_{jx})}{\sum\limits_{k=1}^{3} \exp(V_{k})}.\frac{\exp(V_{jx})}{\sum\
olimits_{l=1}^{n!}\exp(V_{l})}\right)(w_ {x}-C_{v})-C_{b}\right)\end{array}  \tag{11}\]  
where \(w_{x}\) is the parking lot pricing for the x-th operating period. The goal of maximizing revenue from reservation parking can be expressed as follows:  
\[maxZ=\sum\limits_{x=1}^{n}\left(Q\frac{\exp(V_{a})}{\sum\limits_{k=1}^{3}\exp( V_{k})}\left(\sum\limits_{j=1}^{n_{x}}\frac{\exp(V_{a})}{\sum\limits_{k=1}^{3} \exp(V_{k})}.\frac{\exp(V_{jx})}{\sum\
olimits_{l=1}^{n!}\exp(V_{l})}\right)(w_ {x}-C_{v})-C_{b}\right) \tag{12}\]The independent variables in the above profit function are the reservation parking lot price \(w_{x}\) and the traditional parking lot price \(w^{\prime}\). In this case, the reservation parking lot manager as the decision maker can only decide on the price w corresponding to each operation period, i.e., in the objective function of the reservation parking lot party, the decision maker independent variable is only \(w_{x}\). In addition, the utility function in the objective function also has non-decidable parking prices of traditional car parks as well as users' choices.