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Unobserved and observed heterogeneity in risk attitudes: Implications for valuing travel time savings and travel time variability
Transportation Research Part E 112 (2018) 12–18
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Unobserved and observed heterogeneity in risk attitudes: Implications for valuing travel time savings and travel time variability
T
Zheng Li School of Finance, Xi’an Eurasia University, No. 8 Dongyi Road, Yanta District, Xi’an 710065, China
AR TI CLE I NF O
AB S T R A CT
Keywords: Travel time variability Unobserved heterogeneity Observed heterogeneity Risk attitude Expected utility theory Willingness to pay
In this paper, we incorporate the attitude towards risk into a scheduling model to account for travel time variability, using a choice experiment of car commuters choosing from risky alternatives. The parameters that represent unobserved and observed heterogeneity in risk attitudes are jointly estimated within a non-linear utility framework. The model outputs are compared with the results from the model under the assumption of risk attitude homogeneity and we find differences in the willingness to pay for time savings and reduced travel time variability. These findings illustrate that risk attitude heterogeneity plays a role in choice behaviour.
1. Introduction Travel time variability is an important measure of service quality. Bates et al. (1987) classified travel time variability into interday, inter-period and inter-vehicle variability. The research focus is on the impacts of day-to-day travel time variations on travel choice behaviour (Bates et al., 2001; Polak et al., 2008; Ramos et al., 2014). Travel time savings is not sufficient to measure the total benefits of infrastructure improvement projects, in which the benefits of reduced travel time variability or improved reliability should be considered (Ettema and Timmermans, 2006; Fosgerau and Karlström, 2010; Taylor, 2017). OECD (2017) highlighted the significance of including travel time variability in the economic analysis. Several countries such as Australia and New Zealand have already included the value of travel time variability in the cost-benefit analysis for project appraisal (de Jong and Bliemer, 2015). Most travel time variability studies used a linear utility function, which implicitly assumes a risk neutral attitude. This assumption may be appropriate in a deterministic or riskless decision-making environment, for example, there is only one travel time with 100% chance of occurrence for repeated journeys. However, this may not be realistic given that travel time variability is inherent to most transport systems. Given a departure time, there is a chance of arriving early, on time or late (i.e., a travel time distribution). In the presence of travel time variability, the attitude towards risk (risk averse or taking) plays an important role in decision making under risk. Noland and Small (1995) developed a scheduling model in which travel time variability is represented by the expected schedule delay early/late, based on earlier theoretical contributions by Small (1982) and Polak (1987). Polak (1987) also discussed risk-averse and risk-taking behaviour in the context of departure time choice. Using a non-linear scheduling model with an exponential utility specification, Polak et al. (2008) addressed unobserved heterogeneity in risk attitudes. With respect to the non-linear utility specification, this study adopts a power specification to develop the non-linear scheduling model for parameter estimation. In the transport literature, a growing number of studies have incorporated alternative behavioural theories (mainly Expected Utility Theory (EUT), Rank-Dependent Utility Theory (RDUT) and Cumulative Prospect Theory (CPT)) which allow for non-linearity
E-mail address: [email protected]. https://doi.org/10.1016/j.tre.2018.02.003 Received 22 May 2017; Received in revised form 2 February 2018; Accepted 5 February 2018 1366-5545/ © 2018 Elsevier Ltd. All rights reserved.
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in utility, as a way of representing the attitude towards risk (see e.g., Avineri and Prashker, 2003; Michea and Polak, 2006; Schwanen and Ettema, 2009; Dixit et al., 2015). Under RDUT and CPT, a non-linear probability weighting function transforms the probabilities of all possible outcomes (e.g., arriving early, late and on time) into decision weights; while the original probabilities are directly used under EUT. A scheduling model takes into account the consequences of travel time variability (i.e., arriving earlier/later than the preferred arrival time), in which only the probabilities of arriving early and late are used to calculate the expected values of schedule delay early and schedule delay late. Therefore, the non-linear scheduling model of this study is based on EUT. Within this non-linear utility framework, unobserved and observed heterogeneity in travellers’ risk attitudes are jointly estimated, along with the values of travel time savings and travel time variability. In the literature, only Andersen et al. (2012) have empirically addressed both unobserved and observed risk attitude heterogeneity. In this paper, we also investigate the influence of unobserved and observed risk attitude heterogeneity on willingness to pay (WTP), which is the first in the literature to the author’s knowledge. 2. Recent research on travel time variability: alternative behavioural theories This section briefly summarises recent travel time variability studies based on alternative behavioural theories, with the empirical estimation of risk attitude parameters. Amongst the reviewed non-linear utility studies published after 2010, Expected Utility Theory is the predominant framework for understanding travel choice behaviour under risk (Hensher et al., 2011, 2013a, 2013b; Li and Hensher, 2012; Li et al., 2012; Dixit et al., 2013; Wijayaratna and Dixit, 2016; Balbontin et al., 2017), followed by Rank-Dependent Utility Theory (Hensher and Li, 2012; Li and Hensher, 2013; Razo and Gao, 2014; Wang et al., 2014; Dixit et al., 2015) and Cumulative Prospect Theory (Huang et al., 2016; Wen et al., 2017). A common finding of these studies is risk-taking behaviour over travel time variability. Half of them also estimated the WTP values (Hensher et al., 2011, 2013a, 2015; Hensher and Li, 2012; Li and Hensher, 2012; Li et al., 2012; Huang et al., 2016; Balbontin et al., 2017). Six studies accommodated unobserved between-individual heterogeneity in risk attitudes (Hensher et al., 2011, 2013a, 2013b; Li and Hensher, 2012; Li et al., 2012; Li and Hensher, 2013). For observed heterogeneity in risk attitudes, Dixit et al. (2013, 2015) and Wijayaratna and Dixit (2016) tested a number of socioeconomic characteristics, while only Age and Race (African Americans vs. non-African Americans) are statistically significant. None of the existing travel time variability studies addressed both unobserved between-individual heterogeneity and observed heterogeneity in risk attitudes (see e.g., Li et al., 2010; Carrion and Levinson, 2012; Ramos et al., 2014; Shams et al., 2017 for reviews). In this paper, we accommodate unobserved and observed risk attitude heterogeneity in a non-linear scheduling model. An essential component of Rank-Dependent Utility Theory and Cumulative Prospect Theory is non-linear probability weighting which transforms the cumulative probability distribution based on the rank of all possible outcomes for an alternative. However, in a scheduling model, travel time variability is represented by the expected schedule delay early/late (i.e., the minutes of arriving earlier/later than the expected arrival time weighted by their probabilities of occurrence). Under Expected Utility Theory, decision makers compare “the weighted sums obtained by adding the utility values of outcomes multiplied by their respective probabilities” (Mongin, 1997, p.342). Therefore, this study applies Expected Utility Theory to develop the non-linear scheduling model, which x 1−α
adopts the power utility specification (i.e., U = 1 − α ) and directly use the original probabilities of early/late arrival to weight their corresponding outcomes. Expected Utility Theory is a conventional framework for modelling risky choice behaviour. A basic EUT model is given in Eq. (1), with the constant relative risk aversion form estimated as the power specification. Constant absolute risk aversion (CARA) and constant relative risk aversion (CRRA) are two main options for representing the attitude towards risk, where the CARA model form postulates an exponential specification for the utility specification, and the CRRA form is a power specification. CRRA is usually a more plausible description of the attitude towards risk than CARA (Blanchard and Fischer, 1989). 1−α
EU =
∑ m
⎛⎜p x m ⎞⎟ m ⎝ 1−α ⎠
(1)
x m is the mth outcome of an alternative with multiple possible outcomes; pm is the probability associated with the mth outcome; and α is a parameter which needs to be estimated, and the value of (1 − α) reveals the attitude towards risk. In psychological studies, the attribute of x for an alternative is often money, which is a source of utility and its parameter estimate would be positive. The value of (1 − α) is estimated to be less than 1, suggesting a curvature of decreasing marginal utility. However, travel time or variability is a source of disutility, (1 − α) < 1 suggests decreasing marginal disutility. This difference is shown in Fig. 1, which is similar to different risk attitudes in the gain domain and in the loss domain under Prospect Theory. Tversky and Kahneman (1992) provided parametric formulae for the value functions, that is V = x α in the gain domain and V = −λ (−x ) β in the loss domain. The corresponding values estimated by Tversky and Kahneman (1992) are 0.88 for α, 0.88 for β and 2.25 for λ. The estimates of α and β, both less than 1, revealed a risk-averse attitude over monetary gains and a risk-taking (or risk-seeking) attitude over monetary losses. In Fig. 1a, suppose there are two scenarios with an equal expected value of $10: A (sure): winning $10 with 100% chance of occurrence; B (risky): a 50:50 chance of winning $0 or winning $20. If (1-α) < 1, the utility incurred by the sure one would be higher than that incurred by the risky one because of a curvature of decreasing marginal utility. Therefore, the sure one would be preferred and chosen, even with the same expected value of $10. This choice behaviour implies risk aversion. In Fig. 1b, suppose there are two scenarios: A (sure): arriving 10 min later than the preferred arrival time (PAT) with 100% chance of occurrence; B (risky): a 50:50 chance of arriving on time or 20 min later than the PAT. If (1 − α) < 1, the disutility incurred by the sure scenario would be higher than that incurred by the risky one due to decreasing marginal disutility, and hence the risky one 13
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(a): risk averse
(b): risk taking
Fig. 1. decreasing marginal utility (a: money) vs. decreasing marginal disutility (b: time).
would be chosen. This implies risk-taking behaviour. In order to empirically address the attitude towards risk, non-linearity in utility needs to be embedded in a scheduling model. In this non-linear scheduling model, the attitude towards risk associated with these risky attributes (time and variability) is captured through a risk attitude parameter. The overall utility expression for this non-linear scheduling model is given in Eq. (2).
EU = βE PE
L1 − α EΔ1T− α E (T )1 − α + βL PL ΔT + βE (T ) + βCost Cost + βTollasc Tollasc 1−α 1−α 1−α
(2)
EΔT and LΔT are the minutes arriving earlier and later than the preferred arrival time and PE and PL are the probabilities of early arrival and late arrival, used to calculate the expected values; E (T ) is the average travel time. Tollasc is the dummy variable to indicate whether a specific alternative is a tolled road. βE (T ) , βE , βL and βCost, are attribute-associated parameters to be estimated, along with βTollasc, and α is an additional parameter to be estimated. The value of (1 − α) indicating the attitude towards risk ((1 − α) < 1: risk taking; (1 − α) = 1: risk neutral; (1 − α) > 1: risk averse).
3. Empirical application For the empirical application, a stated preference (SP) data set conducted for an Australian toll road project is used, in which each choice task has three alternatives where the first alternative is the revealed preference (RP) alternative – a trip described by its current attribute levels and two SP alternatives that are pivoted around the knowledge base of sampled respondents (i.e., the RP alternative). Travel time variability was represented by a full travel time distribution with three travel scenarios for each alternative per choice task: “arriving x minutes earlier than expected,” “arriving y minutes later than expected,” and “arriving at the time expected, with three corresponding probabilities of occurrence (PE , PL and POn ) to imitate the stochastic nature of travel time variability. The survey firm that collected the data went to great lengths, with the interviewer present, to explain what this meant for each respondent. For example, the 30% associated with a 6-min earlier arrival relative to the expected arrival time (i.e., taking the average travel time of 58 min consisting of 14-min free flow time, 18-min slowed down time and 26-min stop/start/crawling time with 50% chance of occurrence) for Route A in Fig. 2 was explained as ‘for every 10 trips you might take, 3 out of the 10 trips the travel time will be 6 min less than the 58 min stated above as the average time experienced, or a trip time of 52 min’. The vehicle running cost for car travel and any toll cost for the specific trip in question were also included in the SP attributes. In the definition provided to the respondent, running costs include only fuel costs for cars, the most commonly perceived cost for the marginal trip, plus toll costs - the amount of money spent for a specific trip assuming the trip occurred using a toll route. Given the lack of exposure to tolls for many travellers in the study catchment area, the toll levels were fixed over a range, varying from no toll to Au$4.20, with the upper limit determined by the trip length of the sampled trip. The experimental design method of D-efficiency used herein is specifically structured to increase the statistical performance of the models with smaller samples than are required for other less-efficient (statistically) designs such as orthogonal designs. The survey was conducted in Brisbane, Australia, in November 2008. The survey has five major sections: the introduction to the survey task and background on the study, questions describing a current or recent trip in terms of travel times and cost (including tolls if paid), the SP experiment (16 screens), a series of attitudinal questions seeking views on the broader set of quality benefits of toll and freeway roads and socioeconomic questions. For the sampling strategy, a telephone call was used to establish eligible participants from households. The face-to-face interview involves the interviewer entering information into a laptop computer program as the respondent answers a set of questions on each screen. The data is automatically stored in an MS-Access database. The quota for commuters was 300 and 280 commuters were achieved for this study. For the socio-economic profile, the average income of sampled car commuters was Au $67,145 in 2008 and the average age was 42. 14
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Fig. 2. One choice scenario screen (three travel scenarios: early, on-time and late arrival per choice alternative).
4. Model estimation and results A mixed multinomial logit (MMNL) model with a homogeneous risk attitude is estimated as the base model in which the Average Travel Time and time variability (Earlier and Later) are represented by three random parameters. It is then compared with another MMNL model with an additional random parameter applied to Alpha (i.e., unobserved risk attitude heterogeneity) and socioeconomic characteristics representing systematic risk attitude heterogeneity. For the random parameters, only the triangular distribution (unconstrained for Alpha and constrained for time and variability)1 delivered behaviourally meaningful ranges of willingness to pay. The maximum simulated likelihood estimation uses 500 Halton draws. For the MMNL model with a homogeneous risk attitude (Table 1), all estimated parameters are statistically significant at the 95% confidence interval. The estimated parameter for the reference-specific constant (i.e., the constant for the current trip or the RP alternative) is positive, which suggests, after accounting for the captured influences, that sampled respondents, on average, prefer their current trip attribute package relative to the two SP alternatives. Tollasc is negative, which indicates that, on average after accounting for the time and cost of travel, other factors bundled into a ‘toll road quality bonus’ are less desirable for a tolled route than a non-tolled route, mainly due to the lack of exposure to tolls for sampled respondents. Alpha is statistically significant with a tratio of 8.72. The calculated risk attitude value (1 − 0.5698 = 0.4302 < 1), implying that, on average, our sampled car commuters tend to be risk-taking, given that travel time variability is a source of disutility. The willingness-to-pay (per minute) formulae for reduced travel time variability (arriving earlier or later than the PAT) and for savings in the expected or average travel time are given in Eqs. (3a)–(3c).
∂ (U ) ∂ (EΔT )
∂ (U ) = (1−α ) βE PE EΔ(1T− α ) − 1/(1−α )/ βCost = βE PE EΔ−Tα / βCost ∂ (Cost )
(3a)
∂ (U ) ∂ (LΔT )
∂ (U ) = (1−α ) βL PL LΔ(1T− α ) − 1/(1−α )/ βCost = βL PL LΔ−Tα / βCost ∂ (Cost )
(3b)
1 Let c be the centre and s the spread (i.e., half the range). The density starts at c − s, rises linearly to c, and then drops linearly to c+ s . It is zero below c − s and above c+ s . The mean and mode are c. The standard deviation is the spread divided by σ ; hence the spread is the standard deviation times σ . The height of the tent at c is 1/s (such that each side of the tent has area s × (1/s) × (1/2) = 1/2, and both sides have area 1/2 + 1/2 = 1, as required for a density). The slope is 1/s2. For a constrained distribution, the mean parameter is constrained to equal its spread (i.e., βjk = βk + |βk|Tj, and Tj is a triangular distribution ranging between −1 and +1), and the density of the distribution rises linearly to the mean from zero before declining to zero again at twice the mean. Therefore, the distribution must lie between zero and some estimated value (i.e., the βjk). When a constrained triangular distribution is used, the reported standard deviation parameter is the spread parameter.
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Table 1 Non-linear scheduling model with a homogeneous risk attitude (MMNL). Variable
Coefficient
t-Ratio
Non-random parameters Reference constant Cost ($) Tollasc Alpha (α)
0.7045 −0.2867 −0.3095 0.5698
11.98 −16.23 −3.13 8.72
Means for random parameters Average Travel Time (min) Earlier (min) Later (min)
−0.9536 −0.1231 −0.2919
−4.25 −2.17 −7.10
Standard deviations for random parameters Average Time (min) Earlier (min) Later (min) No. of observations
0.9536 0.1231 0.2919 4480
4.25 2.17 7.10
Akaike information criterion (AIC) Rho-squared Log-likelihood
6668.6 0.323 −3327.31
Estimated using Nlogit5.
∂ (U ) ∂ [E (T )]
∂ (U ) = (1−α ) βE (T ) E (T )(1 − α ) − 1/(1−α )/ βCost = βE (T ) E (T )−α / βCost ∂ (Cost )
(3c)
For an average travel time savings being 60 min (E(T) = 60 in Eq. (3c)), the calculated WTP is Au$19.33 (standard deviation = 7.96). On average, each car commuter is willing to pay Au$19.36 (or US$12.76) to reduce one hour’s mean travel time. Suppose, for example, that the probabilities of arriving earlier/later are 0.1 and the minutes of arriving earlier/later are five minutes, the corresponding values of travel time variability are calculated to be Au$0.09 (standard deviation = 0.04) and Au$0.25 (standard deviation = 0.08), i.e., on average, a car commuter is willing to pay Au$0.09 to avoid a 5-min early arrival with a 10% chance of occurrence and Au$0.25 to avoid a 5-min late arrival with a 10% chance of occurrence. The results of the MMNL model with unobserved and observed risk attitude heterogeneity are given in Table 2. To statistically compare the MMNL mode with a homogeneous risk attitude (Table 1) and the MMNL with heterogeneous risk attitudes (Table 2), a likelihood ratio test is used with two degrees of freedom. The calculated test statistic (i.e., two multiplies the difference between the log-likelihood values of two models: −3316.62 and −3327.31 respectively) is 21.38. Given that the corresponding critical value (chi-square) at the 99% confidence level is 9.21, the model with heterogeneous risk attitudes delivers a statistically better fit than the model with a homogeneous risk attitude. Table 2 Non-linear scheduling model with observed and unobserved heterogeneity in risk attitudes (MMNL). Variable
Coefficient
t-Ratio
Non-random parameters Reference constant Cost ($) Tollasc Older Age (Dummy variable: 0.1)
0.7204 −0.3083 −0.3124 0.2113
12.29 −17.33 −3.16 9.82
Means for random parameters Average Time (min) Earlier (min) Later (min) Alpha (α)
−0.5804 −0.1271 −0.3348 0.3806
−3.91 −2.33 −8.06 5.37
Standard deviations for random parameters Average Time (min) Earlier (min) Later (minutes) Alpha (α) No. of observations
0.5804 0.1271 0.3348 0.3484 4480
3.91 2.33 8.06 12.46
Akaike information criterion (AIC) Rho-squared Log-likelihood
6651.2 0.326 −3316.62
Estimated using Nlogit5.
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Fig. 3. The distribution of conditional individual-level mean parameters for risk attitude (1 − α).
Table 3 Comparison between the model with a homogeneous risk attitude and the model with observed and unobserved heterogeneity in risk attitudes. Homogeneous risk attitude (MMNL)
Unobserved and observed heterogeneity in risk attitudes (MMNL)
A single risk-taking attitude value
A distribution of risk-taking attitudes; older being more risk-taking
All car commuters*
Older
Young
60-min mean travel time 5-min earlier (probability = 0.1) 5-min later (probability = 0.1)
19.33 0.09 0.25
12.16 0.08 0.22
28.82 0.12 0.30
Model fit AIC Log-likelihood Source
6668.6 −3327.31 Table 1
6651.2 −3316.62 Table 2
Empirical risk attitude Mean WTP
* 280 car commuters sampled, 16 choice tasks per respondent.
This model (Table 2) reveals significant unobserved heterogeneity in Alpha. The mean Alpha parameter estimates at the individual level for all 280 respondents in the sample is shown in Fig. 3. The purpose of Fig. 3 is to illustrate that Alpha have a distribution at the individual level, rather than a single value of risk attitude estimated in the previous model (Table 1). These reported estimates are based on the individual-specific conditional parameter distributions, and not the unconditional distribution reported in Table 1. The conditioning occurs at the individual level based on the respondent’s choices and attribute levels. These mean estimates are based on 500 repeated draws from the estimated model for the parameters of interest. At the individual level, the value of (1 − α) (i.e., the risk attitude parameter) varies from 0.56 to 0.71, with a mean of 0.61 and standard deviation of 0.03. All individual estimates of Alpha are between 0 and 1 (hence 1 − α < 1), suggesting that all respondents are risk taking. For observed risk attitude heterogeneity, a number of socioeconomic characteristics were tested (e.g., income and gender) but found only Age to be statistically significant. Dixit et al. (2013) investigated structural heterogeneity in risk attitudes including age, income, gender, race, education and location and only Age and Race (African Americans vs. non-African Americans) are statistically significant. This study divided Age into two groups, and Age > 40 is Older Age, given that sampled commuters aged between 24 and 70, with the medium value being 40. The model (Table 2) delivers a positive parameter estimate for Older Age, suggesting that the older respondents tend to be more risk-taking. Two reasons might explain this structural difference. First, they would have more experience of driving so that they could reduce the chance of not arriving on time. Moreover, they might have more flexible arrival times so that the consequence of arriving late would be less serious. For the older respondents, the calculated mean WTP value for an average travel time savings of 60 min is Au$12.16 with a standard deviation of 7.89, and the mean WTP values for avoiding a 5-min early arrival with a 10% chance of occurrence is Au$0.08 (standard deviation = 0.02) and Au$0.22 (standard deviation = 0.05) for avoiding a 5-min late arrival with a 10% chance of occurrence. For the younger respondents, the corresponding WTP estimates are: Au$28.82 with a standard deviation of 18.93, Au$0.12 with a standard deviation of 0.03 and Au$0.30 with a standard deviation of 0.08. The mean WTP values for our sampled older respondent are lower, offset by more risk-taking behaviour. In Table 3, we compare the outputs of two MMNL models with/without risk attitude heterogeneity. In addition to an improvement in model fit, the model allowing for risk attitude heterogeneity delivers different mean WTP values for reducing the mean travel time and travel time variability. These findings reinforce that risk attitude heterogeneity should be taken into account in modelling risky travel choice behaviour.
5. Conclusions The valuation of travel time variability has received considerable attention from researchers and planners. Although travel time variability is embedded in transport systems, the perception of risk is in the mind of the traveller. Travellers may have different 17
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perceptions when experiencing the same level of travel time variability. Therefore, it is important to understand the individual’s attitude towards risk. In this paper, we take into account the endogenous nature of risk by empirically estimating the risk attitude parameter for each sampled car commuter and accounting for the source of systematic risk attitude heterogeneity. The findings show that there is significant heterogeneity in risk attitudes across individuals and the older car commuters tend to be more risk-taking relative to the younger group. We have also investigated the role of risk attitude heterogeneity in the valuation of travel time savings and travel time variability. The identified unobserved heterogeneity and observed heterogeneity in risk attitudes have a notable impact on the WTP for time savings and reduced variability. 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Behav. Sci. 111, 301–310. Wen, C., Wu, W., Fu, C., 2017. Preferences for alternative travel arrangements in case of major flight delays: evidence from choice experiments with prospect theory. Transp. Policy. http://dx.doi.org/10.1016/j.tranpol.2017.02.005. Wijayaratna, K.P., Dixit, V.V., 2016. Impact of information on risk attitudes: Implications on valuation of reliability and information. J. Choice Modell. 20 (3), 16–34.
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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre
Unobserved and observed heterogeneity in risk attitudes: Implications for valuing travel time savings and travel time variability
T
Zheng Li School of Finance, Xi’an Eurasia University, No. 8 Dongyi Road, Yanta District, Xi’an 710065, China
AR TI CLE I NF O
AB S T R A CT
Keywords: Travel time variability Unobserved heterogeneity Observed heterogeneity Risk attitude Expected utility theory Willingness to pay
In this paper, we incorporate the attitude towards risk into a scheduling model to account for travel time variability, using a choice experiment of car commuters choosing from risky alternatives. The parameters that represent unobserved and observed heterogeneity in risk attitudes are jointly estimated within a non-linear utility framework. The model outputs are compared with the results from the model under the assumption of risk attitude homogeneity and we find differences in the willingness to pay for time savings and reduced travel time variability. These findings illustrate that risk attitude heterogeneity plays a role in choice behaviour.
1. Introduction Travel time variability is an important measure of service quality. Bates et al. (1987) classified travel time variability into interday, inter-period and inter-vehicle variability. The research focus is on the impacts of day-to-day travel time variations on travel choice behaviour (Bates et al., 2001; Polak et al., 2008; Ramos et al., 2014). Travel time savings is not sufficient to measure the total benefits of infrastructure improvement projects, in which the benefits of reduced travel time variability or improved reliability should be considered (Ettema and Timmermans, 2006; Fosgerau and Karlström, 2010; Taylor, 2017). OECD (2017) highlighted the significance of including travel time variability in the economic analysis. Several countries such as Australia and New Zealand have already included the value of travel time variability in the cost-benefit analysis for project appraisal (de Jong and Bliemer, 2015). Most travel time variability studies used a linear utility function, which implicitly assumes a risk neutral attitude. This assumption may be appropriate in a deterministic or riskless decision-making environment, for example, there is only one travel time with 100% chance of occurrence for repeated journeys. However, this may not be realistic given that travel time variability is inherent to most transport systems. Given a departure time, there is a chance of arriving early, on time or late (i.e., a travel time distribution). In the presence of travel time variability, the attitude towards risk (risk averse or taking) plays an important role in decision making under risk. Noland and Small (1995) developed a scheduling model in which travel time variability is represented by the expected schedule delay early/late, based on earlier theoretical contributions by Small (1982) and Polak (1987). Polak (1987) also discussed risk-averse and risk-taking behaviour in the context of departure time choice. Using a non-linear scheduling model with an exponential utility specification, Polak et al. (2008) addressed unobserved heterogeneity in risk attitudes. With respect to the non-linear utility specification, this study adopts a power specification to develop the non-linear scheduling model for parameter estimation. In the transport literature, a growing number of studies have incorporated alternative behavioural theories (mainly Expected Utility Theory (EUT), Rank-Dependent Utility Theory (RDUT) and Cumulative Prospect Theory (CPT)) which allow for non-linearity
E-mail address: [email protected]. https://doi.org/10.1016/j.tre.2018.02.003 Received 22 May 2017; Received in revised form 2 February 2018; Accepted 5 February 2018 1366-5545/ © 2018 Elsevier Ltd. All rights reserved.
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in utility, as a way of representing the attitude towards risk (see e.g., Avineri and Prashker, 2003; Michea and Polak, 2006; Schwanen and Ettema, 2009; Dixit et al., 2015). Under RDUT and CPT, a non-linear probability weighting function transforms the probabilities of all possible outcomes (e.g., arriving early, late and on time) into decision weights; while the original probabilities are directly used under EUT. A scheduling model takes into account the consequences of travel time variability (i.e., arriving earlier/later than the preferred arrival time), in which only the probabilities of arriving early and late are used to calculate the expected values of schedule delay early and schedule delay late. Therefore, the non-linear scheduling model of this study is based on EUT. Within this non-linear utility framework, unobserved and observed heterogeneity in travellers’ risk attitudes are jointly estimated, along with the values of travel time savings and travel time variability. In the literature, only Andersen et al. (2012) have empirically addressed both unobserved and observed risk attitude heterogeneity. In this paper, we also investigate the influence of unobserved and observed risk attitude heterogeneity on willingness to pay (WTP), which is the first in the literature to the author’s knowledge. 2. Recent research on travel time variability: alternative behavioural theories This section briefly summarises recent travel time variability studies based on alternative behavioural theories, with the empirical estimation of risk attitude parameters. Amongst the reviewed non-linear utility studies published after 2010, Expected Utility Theory is the predominant framework for understanding travel choice behaviour under risk (Hensher et al., 2011, 2013a, 2013b; Li and Hensher, 2012; Li et al., 2012; Dixit et al., 2013; Wijayaratna and Dixit, 2016; Balbontin et al., 2017), followed by Rank-Dependent Utility Theory (Hensher and Li, 2012; Li and Hensher, 2013; Razo and Gao, 2014; Wang et al., 2014; Dixit et al., 2015) and Cumulative Prospect Theory (Huang et al., 2016; Wen et al., 2017). A common finding of these studies is risk-taking behaviour over travel time variability. Half of them also estimated the WTP values (Hensher et al., 2011, 2013a, 2015; Hensher and Li, 2012; Li and Hensher, 2012; Li et al., 2012; Huang et al., 2016; Balbontin et al., 2017). Six studies accommodated unobserved between-individual heterogeneity in risk attitudes (Hensher et al., 2011, 2013a, 2013b; Li and Hensher, 2012; Li et al., 2012; Li and Hensher, 2013). For observed heterogeneity in risk attitudes, Dixit et al. (2013, 2015) and Wijayaratna and Dixit (2016) tested a number of socioeconomic characteristics, while only Age and Race (African Americans vs. non-African Americans) are statistically significant. None of the existing travel time variability studies addressed both unobserved between-individual heterogeneity and observed heterogeneity in risk attitudes (see e.g., Li et al., 2010; Carrion and Levinson, 2012; Ramos et al., 2014; Shams et al., 2017 for reviews). In this paper, we accommodate unobserved and observed risk attitude heterogeneity in a non-linear scheduling model. An essential component of Rank-Dependent Utility Theory and Cumulative Prospect Theory is non-linear probability weighting which transforms the cumulative probability distribution based on the rank of all possible outcomes for an alternative. However, in a scheduling model, travel time variability is represented by the expected schedule delay early/late (i.e., the minutes of arriving earlier/later than the expected arrival time weighted by their probabilities of occurrence). Under Expected Utility Theory, decision makers compare “the weighted sums obtained by adding the utility values of outcomes multiplied by their respective probabilities” (Mongin, 1997, p.342). Therefore, this study applies Expected Utility Theory to develop the non-linear scheduling model, which x 1−α
adopts the power utility specification (i.e., U = 1 − α ) and directly use the original probabilities of early/late arrival to weight their corresponding outcomes. Expected Utility Theory is a conventional framework for modelling risky choice behaviour. A basic EUT model is given in Eq. (1), with the constant relative risk aversion form estimated as the power specification. Constant absolute risk aversion (CARA) and constant relative risk aversion (CRRA) are two main options for representing the attitude towards risk, where the CARA model form postulates an exponential specification for the utility specification, and the CRRA form is a power specification. CRRA is usually a more plausible description of the attitude towards risk than CARA (Blanchard and Fischer, 1989). 1−α
EU =
∑ m
⎛⎜p x m ⎞⎟ m ⎝ 1−α ⎠
(1)
x m is the mth outcome of an alternative with multiple possible outcomes; pm is the probability associated with the mth outcome; and α is a parameter which needs to be estimated, and the value of (1 − α) reveals the attitude towards risk. In psychological studies, the attribute of x for an alternative is often money, which is a source of utility and its parameter estimate would be positive. The value of (1 − α) is estimated to be less than 1, suggesting a curvature of decreasing marginal utility. However, travel time or variability is a source of disutility, (1 − α) < 1 suggests decreasing marginal disutility. This difference is shown in Fig. 1, which is similar to different risk attitudes in the gain domain and in the loss domain under Prospect Theory. Tversky and Kahneman (1992) provided parametric formulae for the value functions, that is V = x α in the gain domain and V = −λ (−x ) β in the loss domain. The corresponding values estimated by Tversky and Kahneman (1992) are 0.88 for α, 0.88 for β and 2.25 for λ. The estimates of α and β, both less than 1, revealed a risk-averse attitude over monetary gains and a risk-taking (or risk-seeking) attitude over monetary losses. In Fig. 1a, suppose there are two scenarios with an equal expected value of $10: A (sure): winning $10 with 100% chance of occurrence; B (risky): a 50:50 chance of winning $0 or winning $20. If (1-α) < 1, the utility incurred by the sure one would be higher than that incurred by the risky one because of a curvature of decreasing marginal utility. Therefore, the sure one would be preferred and chosen, even with the same expected value of $10. This choice behaviour implies risk aversion. In Fig. 1b, suppose there are two scenarios: A (sure): arriving 10 min later than the preferred arrival time (PAT) with 100% chance of occurrence; B (risky): a 50:50 chance of arriving on time or 20 min later than the PAT. If (1 − α) < 1, the disutility incurred by the sure scenario would be higher than that incurred by the risky one due to decreasing marginal disutility, and hence the risky one 13
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(a): risk averse
(b): risk taking
Fig. 1. decreasing marginal utility (a: money) vs. decreasing marginal disutility (b: time).
would be chosen. This implies risk-taking behaviour. In order to empirically address the attitude towards risk, non-linearity in utility needs to be embedded in a scheduling model. In this non-linear scheduling model, the attitude towards risk associated with these risky attributes (time and variability) is captured through a risk attitude parameter. The overall utility expression for this non-linear scheduling model is given in Eq. (2).
EU = βE PE
L1 − α EΔ1T− α E (T )1 − α + βL PL ΔT + βE (T ) + βCost Cost + βTollasc Tollasc 1−α 1−α 1−α
(2)
EΔT and LΔT are the minutes arriving earlier and later than the preferred arrival time and PE and PL are the probabilities of early arrival and late arrival, used to calculate the expected values; E (T ) is the average travel time. Tollasc is the dummy variable to indicate whether a specific alternative is a tolled road. βE (T ) , βE , βL and βCost, are attribute-associated parameters to be estimated, along with βTollasc, and α is an additional parameter to be estimated. The value of (1 − α) indicating the attitude towards risk ((1 − α) < 1: risk taking; (1 − α) = 1: risk neutral; (1 − α) > 1: risk averse).
3. Empirical application For the empirical application, a stated preference (SP) data set conducted for an Australian toll road project is used, in which each choice task has three alternatives where the first alternative is the revealed preference (RP) alternative – a trip described by its current attribute levels and two SP alternatives that are pivoted around the knowledge base of sampled respondents (i.e., the RP alternative). Travel time variability was represented by a full travel time distribution with three travel scenarios for each alternative per choice task: “arriving x minutes earlier than expected,” “arriving y minutes later than expected,” and “arriving at the time expected, with three corresponding probabilities of occurrence (PE , PL and POn ) to imitate the stochastic nature of travel time variability. The survey firm that collected the data went to great lengths, with the interviewer present, to explain what this meant for each respondent. For example, the 30% associated with a 6-min earlier arrival relative to the expected arrival time (i.e., taking the average travel time of 58 min consisting of 14-min free flow time, 18-min slowed down time and 26-min stop/start/crawling time with 50% chance of occurrence) for Route A in Fig. 2 was explained as ‘for every 10 trips you might take, 3 out of the 10 trips the travel time will be 6 min less than the 58 min stated above as the average time experienced, or a trip time of 52 min’. The vehicle running cost for car travel and any toll cost for the specific trip in question were also included in the SP attributes. In the definition provided to the respondent, running costs include only fuel costs for cars, the most commonly perceived cost for the marginal trip, plus toll costs - the amount of money spent for a specific trip assuming the trip occurred using a toll route. Given the lack of exposure to tolls for many travellers in the study catchment area, the toll levels were fixed over a range, varying from no toll to Au$4.20, with the upper limit determined by the trip length of the sampled trip. The experimental design method of D-efficiency used herein is specifically structured to increase the statistical performance of the models with smaller samples than are required for other less-efficient (statistically) designs such as orthogonal designs. The survey was conducted in Brisbane, Australia, in November 2008. The survey has five major sections: the introduction to the survey task and background on the study, questions describing a current or recent trip in terms of travel times and cost (including tolls if paid), the SP experiment (16 screens), a series of attitudinal questions seeking views on the broader set of quality benefits of toll and freeway roads and socioeconomic questions. For the sampling strategy, a telephone call was used to establish eligible participants from households. The face-to-face interview involves the interviewer entering information into a laptop computer program as the respondent answers a set of questions on each screen. The data is automatically stored in an MS-Access database. The quota for commuters was 300 and 280 commuters were achieved for this study. For the socio-economic profile, the average income of sampled car commuters was Au $67,145 in 2008 and the average age was 42. 14
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Fig. 2. One choice scenario screen (three travel scenarios: early, on-time and late arrival per choice alternative).
4. Model estimation and results A mixed multinomial logit (MMNL) model with a homogeneous risk attitude is estimated as the base model in which the Average Travel Time and time variability (Earlier and Later) are represented by three random parameters. It is then compared with another MMNL model with an additional random parameter applied to Alpha (i.e., unobserved risk attitude heterogeneity) and socioeconomic characteristics representing systematic risk attitude heterogeneity. For the random parameters, only the triangular distribution (unconstrained for Alpha and constrained for time and variability)1 delivered behaviourally meaningful ranges of willingness to pay. The maximum simulated likelihood estimation uses 500 Halton draws. For the MMNL model with a homogeneous risk attitude (Table 1), all estimated parameters are statistically significant at the 95% confidence interval. The estimated parameter for the reference-specific constant (i.e., the constant for the current trip or the RP alternative) is positive, which suggests, after accounting for the captured influences, that sampled respondents, on average, prefer their current trip attribute package relative to the two SP alternatives. Tollasc is negative, which indicates that, on average after accounting for the time and cost of travel, other factors bundled into a ‘toll road quality bonus’ are less desirable for a tolled route than a non-tolled route, mainly due to the lack of exposure to tolls for sampled respondents. Alpha is statistically significant with a tratio of 8.72. The calculated risk attitude value (1 − 0.5698 = 0.4302 < 1), implying that, on average, our sampled car commuters tend to be risk-taking, given that travel time variability is a source of disutility. The willingness-to-pay (per minute) formulae for reduced travel time variability (arriving earlier or later than the PAT) and for savings in the expected or average travel time are given in Eqs. (3a)–(3c).
∂ (U ) ∂ (EΔT )
∂ (U ) = (1−α ) βE PE EΔ(1T− α ) − 1/(1−α )/ βCost = βE PE EΔ−Tα / βCost ∂ (Cost )
(3a)
∂ (U ) ∂ (LΔT )
∂ (U ) = (1−α ) βL PL LΔ(1T− α ) − 1/(1−α )/ βCost = βL PL LΔ−Tα / βCost ∂ (Cost )
(3b)
1 Let c be the centre and s the spread (i.e., half the range). The density starts at c − s, rises linearly to c, and then drops linearly to c+ s . It is zero below c − s and above c+ s . The mean and mode are c. The standard deviation is the spread divided by σ ; hence the spread is the standard deviation times σ . The height of the tent at c is 1/s (such that each side of the tent has area s × (1/s) × (1/2) = 1/2, and both sides have area 1/2 + 1/2 = 1, as required for a density). The slope is 1/s2. For a constrained distribution, the mean parameter is constrained to equal its spread (i.e., βjk = βk + |βk|Tj, and Tj is a triangular distribution ranging between −1 and +1), and the density of the distribution rises linearly to the mean from zero before declining to zero again at twice the mean. Therefore, the distribution must lie between zero and some estimated value (i.e., the βjk). When a constrained triangular distribution is used, the reported standard deviation parameter is the spread parameter.
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Table 1 Non-linear scheduling model with a homogeneous risk attitude (MMNL). Variable
Coefficient
t-Ratio
Non-random parameters Reference constant Cost ($) Tollasc Alpha (α)
0.7045 −0.2867 −0.3095 0.5698
11.98 −16.23 −3.13 8.72
Means for random parameters Average Travel Time (min) Earlier (min) Later (min)
−0.9536 −0.1231 −0.2919
−4.25 −2.17 −7.10
Standard deviations for random parameters Average Time (min) Earlier (min) Later (min) No. of observations
0.9536 0.1231 0.2919 4480
4.25 2.17 7.10
Akaike information criterion (AIC) Rho-squared Log-likelihood
6668.6 0.323 −3327.31
Estimated using Nlogit5.
∂ (U ) ∂ [E (T )]
∂ (U ) = (1−α ) βE (T ) E (T )(1 − α ) − 1/(1−α )/ βCost = βE (T ) E (T )−α / βCost ∂ (Cost )
(3c)
For an average travel time savings being 60 min (E(T) = 60 in Eq. (3c)), the calculated WTP is Au$19.33 (standard deviation = 7.96). On average, each car commuter is willing to pay Au$19.36 (or US$12.76) to reduce one hour’s mean travel time. Suppose, for example, that the probabilities of arriving earlier/later are 0.1 and the minutes of arriving earlier/later are five minutes, the corresponding values of travel time variability are calculated to be Au$0.09 (standard deviation = 0.04) and Au$0.25 (standard deviation = 0.08), i.e., on average, a car commuter is willing to pay Au$0.09 to avoid a 5-min early arrival with a 10% chance of occurrence and Au$0.25 to avoid a 5-min late arrival with a 10% chance of occurrence. The results of the MMNL model with unobserved and observed risk attitude heterogeneity are given in Table 2. To statistically compare the MMNL mode with a homogeneous risk attitude (Table 1) and the MMNL with heterogeneous risk attitudes (Table 2), a likelihood ratio test is used with two degrees of freedom. The calculated test statistic (i.e., two multiplies the difference between the log-likelihood values of two models: −3316.62 and −3327.31 respectively) is 21.38. Given that the corresponding critical value (chi-square) at the 99% confidence level is 9.21, the model with heterogeneous risk attitudes delivers a statistically better fit than the model with a homogeneous risk attitude. Table 2 Non-linear scheduling model with observed and unobserved heterogeneity in risk attitudes (MMNL). Variable
Coefficient
t-Ratio
Non-random parameters Reference constant Cost ($) Tollasc Older Age (Dummy variable: 0.1)
0.7204 −0.3083 −0.3124 0.2113
12.29 −17.33 −3.16 9.82
Means for random parameters Average Time (min) Earlier (min) Later (min) Alpha (α)
−0.5804 −0.1271 −0.3348 0.3806
−3.91 −2.33 −8.06 5.37
Standard deviations for random parameters Average Time (min) Earlier (min) Later (minutes) Alpha (α) No. of observations
0.5804 0.1271 0.3348 0.3484 4480
3.91 2.33 8.06 12.46
Akaike information criterion (AIC) Rho-squared Log-likelihood
6651.2 0.326 −3316.62
Estimated using Nlogit5.
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Fig. 3. The distribution of conditional individual-level mean parameters for risk attitude (1 − α).
Table 3 Comparison between the model with a homogeneous risk attitude and the model with observed and unobserved heterogeneity in risk attitudes. Homogeneous risk attitude (MMNL)
Unobserved and observed heterogeneity in risk attitudes (MMNL)
A single risk-taking attitude value
A distribution of risk-taking attitudes; older being more risk-taking
All car commuters*
Older
Young
60-min mean travel time 5-min earlier (probability = 0.1) 5-min later (probability = 0.1)
19.33 0.09 0.25
12.16 0.08 0.22
28.82 0.12 0.30
Model fit AIC Log-likelihood Source
6668.6 −3327.31 Table 1
6651.2 −3316.62 Table 2
Empirical risk attitude Mean WTP
* 280 car commuters sampled, 16 choice tasks per respondent.
This model (Table 2) reveals significant unobserved heterogeneity in Alpha. The mean Alpha parameter estimates at the individual level for all 280 respondents in the sample is shown in Fig. 3. The purpose of Fig. 3 is to illustrate that Alpha have a distribution at the individual level, rather than a single value of risk attitude estimated in the previous model (Table 1). These reported estimates are based on the individual-specific conditional parameter distributions, and not the unconditional distribution reported in Table 1. The conditioning occurs at the individual level based on the respondent’s choices and attribute levels. These mean estimates are based on 500 repeated draws from the estimated model for the parameters of interest. At the individual level, the value of (1 − α) (i.e., the risk attitude parameter) varies from 0.56 to 0.71, with a mean of 0.61 and standard deviation of 0.03. All individual estimates of Alpha are between 0 and 1 (hence 1 − α < 1), suggesting that all respondents are risk taking. For observed risk attitude heterogeneity, a number of socioeconomic characteristics were tested (e.g., income and gender) but found only Age to be statistically significant. Dixit et al. (2013) investigated structural heterogeneity in risk attitudes including age, income, gender, race, education and location and only Age and Race (African Americans vs. non-African Americans) are statistically significant. This study divided Age into two groups, and Age > 40 is Older Age, given that sampled commuters aged between 24 and 70, with the medium value being 40. The model (Table 2) delivers a positive parameter estimate for Older Age, suggesting that the older respondents tend to be more risk-taking. Two reasons might explain this structural difference. First, they would have more experience of driving so that they could reduce the chance of not arriving on time. Moreover, they might have more flexible arrival times so that the consequence of arriving late would be less serious. For the older respondents, the calculated mean WTP value for an average travel time savings of 60 min is Au$12.16 with a standard deviation of 7.89, and the mean WTP values for avoiding a 5-min early arrival with a 10% chance of occurrence is Au$0.08 (standard deviation = 0.02) and Au$0.22 (standard deviation = 0.05) for avoiding a 5-min late arrival with a 10% chance of occurrence. For the younger respondents, the corresponding WTP estimates are: Au$28.82 with a standard deviation of 18.93, Au$0.12 with a standard deviation of 0.03 and Au$0.30 with a standard deviation of 0.08. The mean WTP values for our sampled older respondent are lower, offset by more risk-taking behaviour. In Table 3, we compare the outputs of two MMNL models with/without risk attitude heterogeneity. In addition to an improvement in model fit, the model allowing for risk attitude heterogeneity delivers different mean WTP values for reducing the mean travel time and travel time variability. These findings reinforce that risk attitude heterogeneity should be taken into account in modelling risky travel choice behaviour.
5. Conclusions The valuation of travel time variability has received considerable attention from researchers and planners. Although travel time variability is embedded in transport systems, the perception of risk is in the mind of the traveller. Travellers may have different 17
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perceptions when experiencing the same level of travel time variability. Therefore, it is important to understand the individual’s attitude towards risk. In this paper, we take into account the endogenous nature of risk by empirically estimating the risk attitude parameter for each sampled car commuter and accounting for the source of systematic risk attitude heterogeneity. The findings show that there is significant heterogeneity in risk attitudes across individuals and the older car commuters tend to be more risk-taking relative to the younger group. We have also investigated the role of risk attitude heterogeneity in the valuation of travel time savings and travel time variability. The identified unobserved heterogeneity and observed heterogeneity in risk attitudes have a notable impact on the WTP for time savings and reduced variability. 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