# Model-Based Dynamic Toll Pricing: An Overview
## 3 Overview of Simulations
### Driver Behavior  
A User Behavior Model reproduces people's choices among different transportation alternatives based on the alternatives' and the users' characteristics. Drivers' choices between two alternatives may be described through binary logit models, where the utility of each alternative is a function of its current travel time and toll [6; 8; 11; 13; 16; 17; 20; 22; 24; 25; 28; 31; 46; 48; 49]. In such cases, general utility function \(U_{j}(t)\) for an alternative \(j\) is formulated as in (12):  
\[U_{j}(t)=\delta\cdot\pi_{j}(t)+\epsilon\cdot\tau_{j}(t)+\zeta, \tag{12}\]  
where \(\delta\) and \(\epsilon\) represent the weighting of travel time and toll price, respectively, and \(\zeta\) is an offset term used to take into account other unobservables. \(\pi_{j}(t)\) and \(\tau_{j}(t)\) are respectively toll price and travel time of alternative \(j\) at time \(t\). Other factors may be taken into account besides toll and travel time, for instance in [14], the utility is also a function of travel time reliability, which, in some cases, has been even found to be more valued than travel time savings [104]. Once defined a utility function, the probability of choosing alternative \(j\) in a binary logit model is expressed by (13):  
\[p_{j}(t)=\frac{1}{1+e^{\Delta U(t)}}, \tag{13}\]  
Figure 5: The distribution of the analyzed studies sorted by traffic simulation method. where \(\Delta U(t)\) represents the difference in the utility function values of the two alternatives at time \(t\). Binary choices between a tolled and an untrolled alternative may also be modeled through the cumulative distribution function of drivers' VOT \(F(\xi)\). The users whose VOT exceeds a particular limit, called the critical VOT \(\xi_{cr}(t)\), are assumed to choose the tolled alternative. The critical VOT is defined as the ratio of the toll price \(\pi(t)\) and the difference in travel times \(\Delta\tau(t)\) between the two alternatives at time \(t\), as in (14):  
\[\xi_{cr}(t)=\frac{\pi(t)}{\Delta\tau(t)}. \tag{14}\]  
Indicating with \(p(t)\) the proportion of drivers choosing the tolled alternative we can formulate the model as in (15):  
\[p(t)=p(\xi_{cr}(t))=1-F(x_{cr}(t)). \tag{15}\]  
Different distribution functions are considered in the literature: a log-normal distribution is proposed in [45]; Weibull distribution is chosen for fitting in [29]; the authors of [12] adopt a simplified variant of the Burr distribution; the authors of [40] employ a discrete distribution of users' VOT; [PERSON] [23] assumes a Gaussian drivers' VOT probability distribution. We note that multinomial logit models may also be considered for modeling choices with more than two alternatives, whereas the driver behavior model described by (14) and (15) cannot be used in such cases. This last model, in turn, can consider drivers' heterogeneity in VOT. Drivers choices can also be modeled through user equilibrium considerations. As mentioned in Section 2.2, the user equilibrium is a Nash equilibrium derived from drivers' selfish choices that operate to pursue travel costs minimization. In [57] the user equilibrium condition between tolled and untrolled alternative for users all characterized by the same VOT is formulated as given by (16):  
\[\tau_{untollat}(t)=\tau_{tollat}(t)+\pi_{tollat}(t), \tag{16}\]  
where \(\tau_{tollat}(t)\) and \(\tau_{untelled}(t)\) indicate the travel time on the tolled alternative and on the untolled alternative at time \(t\), respectively, and \(\pi_{telled}(t)\) is the toll price of the tolled alternative at time \(t\). Condition (16) holds when both alternatives are used; otherwise, only the less expensive alternative is used. Modeling the case of large networks is typically more complex than for two-alternative cases: a route-choice model usually models driver behavior in association with a departure-time-choice model.