Chapter 5 TOLL ROAD CASE STUDY
5.4 Model specifications
5.4.1 Basic model
Issues in model specifications deserve further attentions. Given that this current study accounts for binary route choice, it is naturally to assume that travel time distribution has
22 The floating car data was collected from 4:00 to 10:00 across 11 days, with only 210 observations. It was separately collected by California Department of Transportation and California Polytechnic State University at San Luis Obispo.
23 This is the time taken to travel a 10 mile portion of the study corridor, and the observed range of travel time is from 8 minutes to 20 minutes.
24 A natural question is how to determine the increment of consecutive travel time outcomes. Indeed, it is not necessary to assume 1 minute as the increment, and alternatively travellers may consider more aggregated travel time outcome set or even fuzzy outcome set, for instance, they just simply account for three outcomes, namely the best outcome, normal outcome, and the worst outcome. Whereas, it is impossible to identify the actual outcome set taken into account by travellers, and it is arbitrary to give a random number for the outcome increment. Moreover, as shown in section 5.5.3, the RDEV model with 2 minutes increment gives worse model fit than the model with 1 minute increment. Consequently, 1 minute is selected as the natural increment for travel time outcomes.
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significant influence on travellers’ route choices. Indeed, most relevant studies usually employ mean or median travel time as the variable reflecting the centrality of travel time distribution25. Noticed that expected value of travel time mathematically equals mean travel time, such linear specification is in line with Expected Value Theory (EVT) function. Thus, we treat EVT function as the basic model, and the utility function of choosing the toll road is expressed as:
∑ (5.1)
where is the travel time for the kth outcome of the free road alternative (the travel time for toll road is assumed to be constant at 8 minutes), and is the associated probability (e.g.
a 10 minute travel time outcome with the associated probability). characterizes the ith agent’s specific socio-demographic attributes such as income and gender. , and are the parameters to be estimated. We also tested the inclusion of standard deviation (of travel time) into this model, though the estimated parameter for standard deviation turned out to be not statistically significant (for details refer to Table 5.1).
Existing literature on EUT exhibits a wide range of utility formulations incorporating decision makers’ attitudes toward risk (see subsection 3.3.1 for details). It is arbitrary to select one of them unless we empirically test the actual performances of these utility specifications26. In this chapter, we employ constant relative risk aversion specification to nonlinearly transform utility function. The EUT model specification is expressed as follows:
∑ (5.2)
where the extra parameter characterizes agent’s attitude towards risk. As the traditional method for modelling risky choice behaviour, this EUT specification is capable of explaining whether individuals tend to nonlinearly distort the utility of travel time. In this thesis, EUT also serves as a basic model for the purpose of comparison. The following subsections concentrate on our alternative model specifications.
25 We initially select the centrality-dispersion specification as the basic model structure (it is consistent with Small et al. (2005)). However, it is found that the parameter for standard deviation is not statistically significant at all.
26 A preliminary analysis has been conducted to determine which utility functions better fit the dataset.
Consequently, we found out that either the maximum number of iterations is reached in the exponential specification (e.g. CARA model), or the estimated parameter of travel time is not statistically significant in the power specification (e.g. Box-Cox and CRRA model). This undesirable result may be due to the relatively low quality of travel time data.
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5.4.2 Weighted Utility Theory (WUT) Model
In WUT specification, risky outcome is not only weighted by associated probability, but also by the outcome value per se. Thus, we define the following functional form:
∑ ∑
(5.3)
where is a function assigning an additional weight to risky outcomes, and is the value of the kth travel time outcome. Determining the function of the weight factor in WUT remains an open problem without general solutions. Sugden (2004) interpreted the real-value weight factor function as . We treat the above WUT function as a simple extension of EVT, i.e., each outcome is weighted by ∑
. And EVT is a special case of WUT if is identical or . The basic assumption of WUT is that travellers tend to overweight the outcome with more travel time. Another promising function is the Box-Cox transformation of travel time shown as follows:
{
(5.4)
This is a more general class of power functions which has been widely used in transportation (Mandel, 1998; Lapparent, 2010). In this research, both Sugden’s function and Box-Cox function will be tested.
5.4.3 Subjective Expected Utility Theory (SEU) Model
In this thesis, we adopt the following SEU model to embody the nonlinearity of probability.
∑ (5.5)
where is the weighting function of probability . In line with our EUT model, CRRA utility function is also employed to address travellers’ risk attitudes. As discussed in subsection 4.2.3, there exist a large body of functional forms for . Thus it is worth testing the influence of selecting different weighting functions on the final model fit. It should be
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noted that subjective expected value theory (SEV) theory is a special case of SEU when , and EUT is a special case of SEU when .
5.4.4 Rank-Dependent Expected Utility Theory (RDEU)
The RDEU model is expressed by the following equation:
∑ (5.6)
where characterizes individuals’ decision weights towards risky outcomes, given the outcomes are defined in increasing order, i.e. the outcome with the least travel time (also the best outcome in this context) is ranked as k = 13. We deliberately use in order to discriminate it from of SEU. In RDEU, is numerically determined by the difference between two cumulative subjective probabilities, i.e.
∑ ∑ . Again, a number of functional forms of weighting function have been examined since the 1990s (see Scott, 2006 for details), thus, this research also aims to identify whether the performance of RDEU is affected by choosing weighting functions.
5.4.5 Prospect Theory (PT)
The proposed PT model specification is expressed as follows:
∑ ( ) ∑ ∑ (5.7) where is the reference travel time which, in this case, is assumed to be the same across individuals. And the travel time parameter is divided into gain , loss , and diminishing loss parameters respectively according to the relative location with respect to the reference point . In the space of loss, the sensitivity towards travel time loss is diminishing insofar as travel time outcome turns out to exceed a travel time to be estimated.
The PT model raises the question of how to endogenously estimate the reference point . Surprisingly this issue has not been empirically studied very much despite its critical
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role in PT. In order to solve this nontrivial problem the algorithm shown in Figure 5.1 is applied.
Fix the possible reference point between two consecutive time outcomes , i.e.
where i = 1 to 12 i rp i
t t t 1
Build PT model by setting as the unknown parameter rp
t
Estimate , and etc.rp
t
Determine whether the estimated reference point is
and tt loss( ) tt gain( )
End YES
NO. Switch i to the next outcome.
i rp' i
t t t 1
( ) tt loss ( )
tt gain
Figure 5.1: Algorithm for the estimation of endogenous reference point
To identify the value of , the two-step profile likelihood approach is adopted. Step 1, build the PT models with fixed and candidate ; Step 2, evaluate PT Model 1 with different , and select model which satisfies three criteria, namely better goodness of fit, statistically significant parameter estimates, and reasonable behavioural performances.
Consequently, as shown in the next subsection, in this empirical context all the above criteria are satisfied only when =8.8 min and =13 min.
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