Prospect TheoryReference pointTravel behaviour and heterogeneity

ctingandedict

The literature on behavioural theories shows that travellers employ different criteria in the process of evaluating what

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The Journal of Choice Modelling

The Journal of Choice Modelling 6 (2013) 17331755-5345/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jocm.2013.04.002n Corresponding author. Tel.: +31 15 278 49 14; fax: +31 15 278 31 79.E-mail addresses: g.m.ramos@tudelft.nl (G. de Moraes Ramos), w.daamen@tudelft.nl (W. Daamen), s.p.hoogendoorn@tudelft.nl (S. Hoogendoorn).1 Tel.: +31 15 278 59 27; fax: +31 15 278 31 79.2 Tel.: +31 15 278 54 75; fax: +31 15 278 31 79.the best choice is. The majority of existing route choice models, however, is based on the utility maximization assumptionwhich assumes that people act rationally in order to get the maximum utility (benefit) from the decision made. In the fieldpatterns are even more severely impacted leading to more travel time uncertainty, which is one of the main impafactors on travellers' behaviour (Noland and Small, 1995; Avineri and Prashker, 2003; De Palma and Picard, 2005; HennOttomanelli, 2006). A proper understanding of travellers' behaviour, therefore, is of fundamental importance to prtravellers' decisions and to forecast future traffic conditions on the network.Travel is the result of individual choice behaviour regarding (i) whether to leave home to engage in an activity, i.e.,activity choice, (ii) where to perform the activity, i.e., destination choice, (iii) how to reach the destination, i.e., mode choice,(iv) when to depart, i.e., departure time choice and (v) which route to take, i.e., route choice (Bovy et al., 2006). Altogetherthese travel related decisions directly affect the performance of the transportation network, especially in the scenario ofincreased mobility observed in most major city centres. As a result, travel characteristics such as travel times and congestion1. Introductionuse as a reference to distinguish gains and losses in the experienced travel times. Moreover,the question can be asked whether all travellers have the same reference point or whetherheterogeneity in their behaviour plays an important role.

This paper aims to (i) provide a behavioural interpretation of the reference point,(ii) investigate the role of heterogeneity in the reference point and (iii) discuss how to takeheterogeneity into account. These aspects are discussed with the aid of an empirical routechoice experiment and a model specification in which travel time is the main variable. Twomodel frameworks are proposed, one accounting for heterogeneity and another consideringno heterogeneity in travellers' behaviour, and their outcomes are compared.

Results show improvements in the ability of Prospect Theory to predict route choicebehaviour by accounting for heterogeneity in the reference point. This is particularly thecase when the reference point reflects travelers' route preferences. Statistical analyses showthe significance of accounting for heterogeneity in travellers' behaviour. Thus, we cannotreject the hypothesis that heterogeneity leads to improvements in the prediction ability ofProspect Theory.

& 2013 Elsevier Ltd. All rights reserved.Modelling travellers' heterogeneous route choice behaviouras prospect maximizers

Giselle de Moraes Ramos n, Winnie Daamen 1, Serge Hoogendoorn 2

Department of Transport & Planning, Faculty of Civil Engineering and Geosciences, Delft University of Technology, P.O. Box 5048, 2600 GA Delft, The Netherlands

a r t i c l e i n f o

Available online 28 May 2013

Keywords:

a b s t r a c t

The use of Prospect Theory to model route choice has increased in the past decades. Themain application issue is how to define the reference point, i.e., the value that travellers

As both intuition and literature suggest, attitudes towards risk vary across the population. As a result, travellers may have

(i)

G. de M. Ramos et al. / Journal of Choice Modelling 6 (2013) 173318(ii) The reference point reflects travellers' (risk) preferences when making route choice decisions, i.e., the reference point isaligned with the travel time distribution of the preferred route.

(iii) In case pre-route information is provided, such as travel time, travellers might use that value as a reference point.

Different fromwhat has been observed in the literature, the contributions of this paper lie on the investigation of the referencepoint in light of its behavioural appeal and the on the role of heterogeneity in the reference point. As a result, we try to graspthe rationale behind risky and risk-averse choices throughout the reference point and the relationship between travellers'heterogeneous preferences and the reference point. This is done by directly applying Prospect Theory in two model specifications,one accounting for heterogeneity and another considering no heterogeneity. Afterwards, their results are compared.

The model comparison is based on data from an empirical route choice experiment (Bogers, 2009) in which travellerswere asked to make choices among three possible routes: route 1 consisting mainly of highways, route 2 consisting mainlyof rural roads and route 3 consisting partly of highway and partly of urban roads. Two conditions of information provisionand two travel purposes were investigated which resulted in four scenarios.

Rather than claiming a higher importance of heterogeneity in the reference point to other types of heterogeneity related torisk, we intend to demonstrate, based on empirical data, the benefits of taking the heterogeneity in the reference point intoaccount. Despite the existence of a great variety of models, (mostly) based on the utility maximization assumption, that areable to describe route choice behaviour quite well, we propose to investigate the suitability of Prospect Theory due to itspotential to better capture travellers' behaviour. Results from previous research conducted by the authors suggest thesuitability of Prospect Theory to model route choice behaviour and that depending on the reference point Prospect Theory canperform better than Utility Theory (Ramos et al., 2011). This motivated further investigation into the role of heterogeneity inthe reference point, which is presented here.

Another well-known factor to be observed is that provision of travel information is likely to change the level ofuncertainty of travel related decisions. Modelling its impact is hardly a simple task. Additional information together withadvanced technologies, such as GPS-based path-finders, are likely to contribute to reduce travel time uncertainty, to enabletravellers to choose efficiently among available routes, to save travel time and to reduce congestion (European Commission,2008). The impact of information, however, is likely to be sensitive to travellers' behavioural and cognitive response toinformation that is much less understood. In addition, effects such as experience and learning also seem to play animportant role in the decision making process (Ben-Elia and Shiftan, 2010).The reference point varies among travellers and over time, i.e., in case travellers' route preferences change over time,such as switching to a more reliable route instead of a fast but unpredictable route, the reference point will followtravellers' new behaviour.inves

ure travellers' perceptions of travel time variability and how this influences their route choices over time, we propose totigate travellers' heterogeneity in relation to their reference point. We hypothesise the following:diffecaptrent perceptions on how to characterise outcomes into gains or losses and thus different reference points. In order toof route choice under uncertainty Expected Utility Theory (Von Neumann and Morgenstern, 1947), in particular, is the mostwidely used theory (De Palma and Picard, 2005). Situations dealing with routes' travel times definitely involve uncertainty.For instance, irrespective of the experience someone has gathered due to frequently driving a specific route, the travel timesmay reasonably vary depending on the traffic conditions.

Despite the widespread use of Expected Utility Theory, experiments in behavioural studies have found deviations fromits axioms leading to the development of Non-Expected Utility Theories of which Prospect Theory (Kahneman and Tversky,1979) is the most discussed (Starmer, 2000). Prospect Theory argues that choices are based on gains and losses measuredagainst a reference point, i.e., values above it are perceived as gains and values below it as losses. What matters, therefore, isthe relative gain and not the final state of wealth or welfare as argued by Expected Utility Theory.

Prospect Theory has been widely used in the field of economics, but applications in the field of transport are relativelyrecent (Sumalee et al., 2005; Avineri, 2006; Connors and Sumalee, 2009; Gao et al., 2010). Its application has been facingtwo main issues: (i) definition of meaningful reference points within the travel behaviour context and (ii) estimation ofappropriate parameters for the value and weighting functions.

The lack of consensus about the meaning of the reference point in a route choice context has often surfaced in theliterature. For instance, De Palma et al. (2008) suggest that the determination of reference points is one of the majorobstacles for the application of Prospect Theory and that it is likely that the reference point varies depending on individualsand choice contexts. While for situations dealing with monetary outcomes, zero is the usual reference point (meaningneither gains nor losses), for situations dealing with travel times this value may vary, for instance, with respect to thedecision maker, to the distances travelled, to the level of stress and to constraints regarding arrival time (Schwanen andEttema, 2009; Senbil and Kitamura, 2004). Therefore, questions concerning the meaning of the reference point in situationsinvolving route choice as well as its pattern over time are raised. In other words, it still has to be made clear what valuestravellers use as a reference to distinguish experienced travel times into gains and losses. Do all travellers have the samereference point or does heterogeneity in their behaviour play an important role? How do repetitive route choices influencethe reference point over time?

Although this paper focuses on the role of heterogeneity in the reference point rather than on the impact of informationon travellers' behaviour, given that two conditions of information provision are considered, we expect that travellers showdifferent route choice behaviour. For instance, in the scenarios in which no information was provided it is expected thattravellers are more willing to explore the routes in order to get familiar with their characteristics. As a result, higherswitching/exploration rates should be observed. As time evolves and travellers get familiar with the travel times of theroutes, routes' switching rates should decrease. In the scenarios in which information was provided, determining a routeexploration pattern is not straightforward because it also depends on the compliance rate with the information. We expect,however, that exploration rates of travellers that in general do not comply with information to be similar to the situation inwhich no information was provided. For travellers that in general comply with the information, as the information providedwas very accurate, we expect that as time evolves and travellers become more confident about the reliability of theinformation, compliance rates with the information should gradually increase.

G. de M. Ramos et al. / Journal of Choice Modelling 6 (2013) 1733 19The next section presents main distinctions between the fundamentals of Prospect Theory in relation to Expected UtilityTheory and a discussion about the role of heterogeneity and the reference point. In Section 3, the proposed case study ispresented and in Section 4 two model specifications are introduced. Comparisons between the model specifications arepresented in Section 5. Finally, conclusions and recommendations for future research are provided in Section 6.

2. Behavioural foundations of Prospect Theory

Prospect Theory, a descriptive model of decision making under risk, was developed in 1979 as a critique to ExpectedUtility Theory based on the main assumption that choices are based on gains and losses measured against a reference point(Kahneman and Tversky, 1979). The point of the critique is that due to the reference point, changing the ways in whichoptions are presented generates predictable shifts in preferences and systematically violates the axioms of Expected UtilityTheory, in particular the independence axiom. The predictable shifts in preferences are not necessarily categorized as good orbad, but instead reflect how people behave when making choices.

2.1. Initial developments

Considering that a prospect is a consequence associated to a probability of occurrence, the independence axiom states that forall prospects x, y, z: if xy then (x, p; z, (1p))(y, p; z, (1p)). This implies that the preference between two prospects isindependent of the common components. Consequently, in the evaluation of a prospect their common components are perceivedas if they could be cancelled (Weber and Camerer, 1987). Experiments such as the Allais Paradox (Allais, 1953) presented below haveshown that the independence axiom is not necessarily applicable and that due to both the framing of options and the referencepoint, shifts in preferences can be observed. Contrary to the expected utility assumption that people behave as utility maximizersand have well defined preferences, Prospect Theory argues that preferences vary depending on how alternatives are presented.

The Allais Paradox is the first famous counter example to Expected Utility Theory to show shifts in preferences. Allais proposedtwo experiments in which people had to choose between two lotteries in each of them (Table 1). A person with expected utilitypreferences would choose option b in both experiments because the independence axiom requires that the difference between thepairs of lotteries to be ignored. Thus, the common consequence of 0.89 chance of winning $1,000,000 in the lotteries a1 and b1, ora 0.89 chance of (winning) $0 in the lotteries a2 and b2 would be cancelled and in both experiments the utility of the choice (u)would be 0.1 u($5M)40.11 u($1M). Allais, however, expected that when faced with these types of choices people would opt for a1in the first situation (win for certainty) and b2 in the second (very different rewards for slightly different winning probabilities). Theexperiments performed confirmed this intuition, and consequently violated the independence axiom.

Following Allais, Prospect Theory was further developed by explaining choice behaviour as a two-step process: an initialphase of editing and a subsequent phase of evaluation. In the editing phase the options are organized and reformulated bythe application of heuristics to simplify the evaluation. In the evaluation phase the prospect is subjectively valued. Theevaluation phase is divided into two scales: a weighting function (p) and a value function v(x). The probability-weightingfunction (p) associates to each probability of occurrence p a decision weight that reflects the impact of p on the overallvalue of the prospect. The value function v(x) reflects the subjective value of the outcome, thus measuring the deviationsfrom the reference point. Thus, neither v(x) is perceived as valuing x nor (p) as valuing p.

Table 1The Allais Paradox.

Experiment 1 Experiment 2

Lottery a1 (to win) Lottery a2 (to win)$1,000,000 with certainty $1,000,000 with probability 0.11

$0 with probability 0.89

Lottery b1 (to win) Lottery b2 (to win)$5,000,000 with probability 0.1 $5,000,000 with probability 0.1$1,000,000 with probability 0.89 $0 with probability 0.9$0 with probability 0.01

G. de M. Ramos et al. / Journal of Choice Modelling 6 (2013) 1733202.2. Cumulative Prospect Theory

In 1992 advances were made to extend Prospect Theory to uncertain outcomes (Tversky and Kahneman, 1992) which led to the

Value

Outcome

Reference Point

Gains Losses

Fig. 1. Illustration of the shape of the value function.

p (probability)1 0

(p)

(p)1

Weighted probability

Fig. 2. Illustration of the shape of the weighting function.development of Cumulative Prospect Theory. This extension was necessary to allow evaluation of situations involving uncertainty inwhich some of the outcomes or probabilities are unknown. This is, for instance, the case for situations involving travel times.

Cumulative Prospect Theory employs cumulative rather than separable decision weights. It is applicable to uncertain andrisky prospects and allows different weighting and value functions for gains and losses due to people's different perceptionof gains and losses. Distinct functions associated to positive and negative outcomes were proposed and defined in terms ofspecific parameters that enable capturing choice behaviour (Figs. 1 and 2).

The value function is defined by three parameters, , and which are responsible for respectively reflecting (i) the degree ofdiminishing sensitivity for gains, (ii) the degree of diminishing sensitivity for losses, i.e., the decrease on the marginal value of gainsand losses with their magnitude, and (iii) the degree of loss aversion, i.e., the aggravation one experiences when incurring losses.

The weighting function, on the other hand, is defined in terms of two parameters, and , that capture the distortionin the perception of probabilities for gains and losses. Its reversed s-shape is derived from (i) the overweight of smallprobabilities, meaning that people are risk-seekers when offered low-probability high-reward prizes and (ii) the under-weight of moderate and high probabilities, implying that (a) people are relatively insensitive to probability difference in themiddle of the range and (b) the prevalence of risk aversion in choices between probable gains and certainty; and riskseeking in choices between probable and sure losses. In addition, the curve of the weighting function starts in zero andfinishes in one ((0)0 and (1)1) and it is asymmetrical with the inflection point at about 0.3 (Prelec, 1998). Theaforementioned parameter specifications for the value and weighting functions are based on their original formulation(Tversky and Kahneman, 1992). For the weighting function in particular, alternative formulations such as the power function(Prelec, 1998) often appears in the literature (De Palma et al., 2008).

Given the distinct functions for positive and negative outcomes, V+ and V respectively, the overall value of a prospect isdefined by VV++V:

V n

i 0i pvxi and V

0

i mi pvxi 1

where min and m and n are the number of negative and positive outcomes respectively.Considering that a prospect f(xi, Ai) is given by a probability distribution p(Ai)pi, the probabilistic or risky prospect

can be viewed as (xi, pi), and the decision weights are defined by

n pn; m pm

G. de M. Ramos et al. / Journal of Choice Modelling 6 (2013) 1733 21i pi pnpi1 pn for 0in1i pm pipm pi1 for 1mi0 2

In order to contribute to a better understanding and application of the above, consider a gamble with eight playing cardsfrom ace (one) to eight. You have to pick one card from the pack x1,, 8, check the result, put the card back and then do itagain. If x is odd, you pay $x, otherwise you receive $x. Considering equivalently probable outcomes, f yields the outcomes(7, 5, 3, 1, 2, 4, 6, 8), each with probability 1/8. Therefore, f+(2, 1/8; 4, 1/8; 6, 1/8; 8, 1/8) and f(7, 1/8; 5, 1/8; 3,1/8; 1, 1/8). Using Eq. (2),

V V V v21=23=8 v43=81=4 v61=41=8v81=80 v71=80 v51=41=8v33=81=4 v11=23=8

The weighting function is given by

p p

p 1p1=and p p

p 1p1=3

where and define the curvature of the weighting function.The value function is given by

vx x if x0 if xo0

(4

where x is the attribute to be measured against the reference point, reflects the degree of loss aversion and and measure the degree of diminishing sensitivity.

The values of these parameters as originally estimated for situations involving monetary outcomes are 0.88,2.25, 0.61 and 0.69 (Tversky and Kahneman, 1992). These parameters have also been estimated in a travelbehaviour context of employed parents coping with unreliable transport networks when picking up their children fromchildcare at the end of the workday (Schwanen and Ettema, 2009). It was assumed that both the value and weightingfunctions were similar for gains and losses, i.e., and . Results show that 1.09, 1.10, 1.271.37 and 0.82,0.84 revealing some attenuation of the properties of Prospect Theory, in particular the loss aversion effect () and thedistortion in the perception of probabilities ( and ). For a comprehensive review of Prospect Theory, we refer to Kahnemanand Tversky (1979) and Tversky and Kahneman (1992).

2.3. Evaluating travellers' route choice behaviour as prospect maximizers and the role of the reference point

The use of Prospect Theory to model route choice behaviour is relatively recent and studies have been focusing onthe effects of travel time uncertainty on equilibrium conditions (Sumalee et al., 2005, 2009; Avineri, 2006; Connors andSumalee, 2009; Gao et al., 2010). In general, comparisons between Prospect Theory and Expected Utility Theory basedequilibrium conditions are shown with the aid of illustrative examples and, despite the emphasis put on the role/importance of the reference point and other parameters, results are usually preliminary insights into the differencesbetween these equilibrium conditions. This is mainly due to the lack of consensus about the behavioural meaning of thereference point.

In situations involving route choices the reference points that are usually adopted are based on the travel times of theproposed case studies, on average travel times to work as reported by transport studies (30 min) or free-flow travel time.However, as discussed in Van de Kaa (2010), when commuters are confronted with alternative routes with different traveltime distributions, the reference point might encompass the mean travel time of the preferred route, the distribution overtime and possibly also a notion of an acceptable bandwidth of arrival times and a preferred departure time. In addition, asmany studies suggest, the adopted reference point is influenced by implicit information, explicit information, or evenirrelevant information (White et al., 1994; Kristensen and Grling, 1997).

The behavioural appeal of the reference point is presented in some studies, such as De Borger and Fosgerau (2008),in which the reference travellers might have in mind when making route choices is discussed. Outcomes suggest theimportance of recent experiences in the definition of the reference point. Connection between the reference point andtravellers' risk preferences, however, has not been found in the literature.

2.4. Role of heterogeneity in the reference point

Another aspect that has not been tackled in the literature is the heterogeneity in the reference point. Experiments inpsychological and social studies have shown that due to people's different perceptions of gains and losses, the perception ofuncertainty and probabilities systematically vary (Wu and Gonzalez, 1996). A similar effect is observed in repeated choicetasks in which some people may keep on updating their reference while others might systematically choose the mostrewarding probabilistic alternative after a learning period (Schul and Mayo, 2003). Despite the importance of heterogeneity

as discussed in the literature and shown in travellers' behaviour analysis involving discrete choice models (Polak et al., 2008;Gopinath, 1995) and car following models (Ossen and Hoogendoorn, 2011), it is usually argued that a single reference pointis used (Zhang et al., 2010; Avineri, 2009) which implies that all travellers value gains and losses similarly. In reality,however, there might be two or more reference points (Avineri and Bovy, 2008; Van de Kaa, 2008) which may vary not onlyamong travellers, but also over time.

Over the past decade considerable progress has been made in the characterisation of unobserved taste heterogeneity intravel choice behaviour (Hess, 2007; Koppelman and Sethi, 2005; Srinivasan and Mahmassani, 2003; Arnott et al., 1992). The

G. de M. Ramos et al. / Journal of Choice Modelling 6 (2013) 173322term taste heterogeneity refers to situations in which different decision makers take into account either different factors orthe same factors in different ways than other factors. Two main approaches have been used to represent the variations inpreferences among individuals (Ben-Akiva and Lerman, 1985). The most common is the partition of the data into a fixednumber of segments usually using demographic characteristics (exogenous segmentation). Alternatively, behaviourallyhomogeneous segments are directly identified from the data (endogenous segmentation).

Route choice behaviour in particular has been widely investigated in the presence and absence of information and itappears that travel time is one of the most relevant factors influencing decisions. Due to traffic congestion, however, traveltimes are uncertain and travellers are likely to have different perceptions of the benefits of choosing a specific route inrelation to another. As a result, it is natural that they also have different reference points regarding perceptions for gains or alosses. For instance, while for a more conservative traveller a 40 min travel time between A and B may be considered good,thus perceived as a gain, for a more aggressive traveller, or someone running late, the same travel time may be perceivednegatively and thus as a loss.

Despite evidence that route choice behaviour varies with drivers' familiarity and prior experience, availability andperception of alternative routes, travel times, delays, incidents and availability of information (Srinivasan and Mahmassani,2003), empirical exploration of variations in reference points across respondents are scarce (Hess et al., 2011). In addition, asdiscussed in Van de Kaa (2010), the large number of studies investigating interpersonal heterogeneity in application ofchoice behaviour strategies have shown a better descriptive ability than homogeneous compensatory attribute processes.Therefore, in a spatially dynamic framework, such as the one proposed in this paper (in which travellers make daily routechoices among three alternative routes), it becomes particularly important to account for heterogeneity as this helpscapturing individual preferences that persist over time.

3. Case study: investigating travellers' route choice behaviour

This case study is based on the data from the experiment performed by Bogers (2009), in which a travel simulator wasused to investigate travellers' route choice behaviour. The experiment simulated daily departures at 8:00 a.m. and arrivalswithin 1 h at the destination for either a meeting with colleagues or a job interview. Travellers were recruited from thedatabase of the Dutch Organization of Road Users (ANWB) and as an incentive to join the experiment one of the travellerswould be awarded a navigation system. The prize was randomly drawn among the participants without specifying a goalsuch as minimizing travel times or arriving on time (which would potentially influence their behaviour). The participantsmade 40 consecutive choices among three routes with an approximate length of 30 km. These routes had the followingcharacteristics: (i) route 1 consisted mainly of highways, it was the fastest route, (ii) route 2 consisted mainly of rural roads,it was the most reliable route and (iii) route 3 consisted partly of highways and partly of urban roads (going through the citycentre), it was the route of intermediate performance (Fig. 3).

Two conditions of information provision were investigated in this case study: (i) no information provision and(ii) provision of travel time in minutes. Information was provided at the beginning of every new trip and only afterreviewing this information travellers could decide which route to take. Changing routes after the journey had started wasnot possible. The travel information was provided for all three routes and based on the true travel times, thus it was veryaccurate for all routes. The information deviated at most 2 min in 85% of the days and approximately between 10 and 30 minin the remaining 15% of the days. In addition, the information provided usually underestimated the true travel times.Feedback information with respect to the true travel time of the chosen route was provided at the end of every journey.Combination of the two travel purposes, i.e., meeting with colleagues and job interview, and the two conditions ofinformation provision, i.e., no information and information in minutes, resulted in four scenarios to be investigated in thiscase study.

An important aspect to be observed is that no prescriptive advice to choose a specific route was provided. Insteadinformation regarding the travel times of each route was provided. However, as the information was very accurate, we

Route 1: highways

Route 2: rural roads

Route 3: highways + urban roads

Origin Destination

Fig. 3. Illustration of the characteristics of the routes in the experiment.

G. de M. Ramos et al. / Journal of Choice Modelling 6 (2013) 1733 23assumed that travellers would choose the fastest route, i.e., the route recommended to be the fastest. The travel timedistributions of each route and characteristics of the scenarios investigated are respectively depicted in Tables 2 and 3.

We assumed that during the initial 10 route choices travellers were getting experience about the routes' travel times. Asof the 11th day, we applied Prospect Theory to predict travellers' route choice behaviour. The participants, however, werenot informed that the initial 10 route choices counted as experience period. The idea behind the considered experienceperiod was to enable travellers to get to know what to expect from the travel times of each route. Then, as of the 11th day,(in theory) travellers would start making choices following a specific route choice behaviour instead of random routechoices.

4. Model specication

We focus our investigation on the behavioural meaning of the reference point and on its influence on the ability of ProspectTheory to capture travellers' behaviour. The influence of other parameters was not taken into account. As a result, the

Table 2Travel time distributions for each route.

Route Description Mean traveltime (min)

Variance of traveltime (min)

10th percentile(min)

90th percentile(min)

1 30 Draws: Gumbel distribution (35, 1)a 10 draws: Gumbeldistribution (70, 1)

44 233 34 70

2 All draws: Gumbel distribution (53, 1.25) 53 1 52 543 All draws: normal distribution (47, 12) 47 146 33 56

a The resulting travel times were randomly distributed over the 40 days.

Table 3Scenarios investigated and number of travellers in each of them.

Travel purpose Condition of information provision

No information Travel time in minutes

Meeting with colleagues Scenario 1 Scenario 3(20 travellers) (42 travellers)

Job interview Scenario 2 Scenario 4(25 travellers) (46 travellers)parameters of the value and weighting function are set to the originally estimated values, i.e., 0.88, 2.25, 0.61 and0.69 (Tversky and Kahneman, 1992). Nevertheless, as discussed in Van de Kaa (2010), despite the attempts to estimateparameters for the travel behaviour context, the originally estimated parameters might still offer the best functionaldescription of travel choice under uncertainty. Therefore, in spite of not using specific parameters for the travel behaviourcontext, the values adopted are so far considered to be the best estimates.

We propose a simple model specification in which travel time is the only attribute when no information was provided,otherwise the travel information is the only attribute of the model. By a direct application of Prospect Theory, the predictedroute was determined based on the maximum prospect theoretical value (MaxPT) of travel times or information providedand then compared with the route chosen by the travellers. The prediction ability of Prospect Theory was determined by thenumber of correct predictions for each traveller in the experiment.

4.1. Considering no heterogeneity

Under the no heterogeneity model specification, we assume that all travellers have the same reference point and this valueis updated as travellers get more experience. During the experiment participants made daily route choices among threeroutes and we assume that after each route choice travellers' expectations, and consequently their reference point, wereupdated. In addition, after each trip the travel time distributions were updated taking the new travel times into account. As aresult, for choices made on the 11th day, data from the initial 10 days was taken into account. For choices made on the 12thday, data from the initial 11 days was taken into account and so on.

We propose the following reference points for the situations in which no informationwas provided: (i) the mode of traveltimes of the fastest route, (ii) the mode of travel times of the most reliable route, (iii) the average travel time of all routes and(iv) the minimum travel time of the fastest route. In all cases, the reference point is based on the actual travel times, thus onthe draws. The first reference point implies that travellers derive satisfaction from getting the best probable result; thesecond assumes that travellers are more conservative in their choices and tend to avoid losses, therefore behaving risk

G. de M. Ramos et al. / Journal of Choice Modelling 6 (2013) 173324averse; the third reference point does not have a clear behavioural meaning and aims to explore the type of choice predictedunder this circumstance; the fourth reference point is a specific group within the first and in behavioural terms mean thattravellers are (very) aggressive in their choices and only want the best possible route.

For the situations in which travel time information was provided, similar reference points were defined. However, insteadof referring to the travel times of the routes, they refer to the travel information. The consequence of this is that theweighting function reflects probabilities of occurrence of specific travel information instead of probabilities of occurrence oftravel times. Similarly to the situations in which no information was provided, the reference point is based on the actualinformation provided. In addition, a fifth reference point, the travel information itself, i.e., the predicted travel time of thefastest route, is also proposed in order to investigate whether travellers update their reference based on the information.Besides this, it is also investigated whether people base their decisions on the real travel times and use the informationsolely as a reference, thus in the value function only. Under this condition the MaxPT is determined based on the real traveltimes, and not on the travel information and a sixth reference point, travel information_2, is evaluated. Under the proposedsixth reference point, when evaluating the weighting function we refer to probabilities of occurrence of travel times andwhen evaluating the value function we refer to the travel information.

The MaxPT is determined based on Eqs. (3) and (4). In the value function, x measures the deviation from the referencepoint (RP) and values xRPTT or RP-info depending on whether information is provided. TT is the real travel time and infothe travel information of a specific route. In the weighting function, p is the probability of occurrence of travel times or travelinformation.

4.2. Accounting for heterogeneity

Under the model specification in which heterogeneity is taken into account, we argue that travellers' route choicebehaviour varies across the population. As a result, each traveller has its own reference point depending on its behaviouralcharacteristics. In addition, similar to the situation in which no heterogeneity is taken into account, we assume thattravellers update expectations and consequently also update their reference point after each route choice.

In order to reflect travellers' usual behaviour, and thus capture risky and conservative behaviours, when no informationwas provided we propose that the reference point on day d is defined as: (i) the travel time of the most frequently chosenroute up to the previous day and (ii) the travel time of the most frequently chosen route in the previous 5 days. Wheninformation was provided, the reference point on day d is defined as (i) the travel information of the most frequently chosenroute up to that day and (ii) the travel information of the most frequently chosen route in the previous five days.For situations involving provision of information, the specifications of the value and weighting functions follow bothaforementioned definitions for the travel information and for the travel information_2, i.e., the weighting function accountsfor both the probabilities of occurrence of travel times and for travel information.

The main difference between the condition in which travellers' heterogeneity is considered and the condition in which itis not considered is that in the first case each traveller is considered as a unique individual. As a result, each traveller has itsown behavioural characteristics. The consequence of this is that we are able to model travellers' preferences individually,considering their preferences in all days of the experiment as well as 5 days prior to the choice made (and other situationsthat could have been defined). The reference point is the element responsible to capture travellers' preferences in allsituations. Conversely, when no heterogeneity is considered, all travellers are assumed to behave similarly. Therefore, onlytheir general behaviour can be captured, i.e., the behaviour of the majority. As a result, it does not make sense to propose areference point that reflects short term switches in preferences such as condition (ii) above.

5. Results and discussion

The results presented in this section focus on two main aspects: travellers' behaviour and the role of the reference pointin the prediction ability of Prospect Theory. With respect to travellers' behaviour, we discuss their route choice behaviourwith respect to different sources of information. On the role of the reference point in the prediction ability of ProspectTheory, we discuss the added value of accounting for heterogeneous reference points and aligning it to travellers' behaviour.These aspects are discussed in Sections 5.1 and 5.2 respectively.

5.1. Travellers' behaviour in relation to information

Travellers' route choice behaviour in scenario 1 (meeting with colleagues+no information) show that during the experienceperiod, i.e., during the initial 10 days, routes 1, 2 and 3 were respectively chosen 62%, 18.5% and 19.5% of the times. After thisperiod, i.e., between the 11th and the 40th day, route 1 was chosen 40.5% of the times, route 2 was chosen 48.8% and route 3 waschosen 10.7%. This indicates a sharp decrease in the preference of both routes 1 and 3 and a corresponding increase in thepreference of route 2. Implication of this is that after getting experience about the routes' travel times, travellers tend to be moreconservative (risk averse) in their choices, thus preferring the most reliable route.

In scenario 2 (job interview+no information), the percentage of travellers choosing the most reliable route in the experienceperiod was higher than in scenario 1. During the experience period, route 1 was preferred 50.9% of the times, route 2 was preferred28.8% and route 3 was preferred 20.4%. During the remaining days, route 1 was chosen 34.5% of the times, route 2 was chosen

52.0% of the times and route 3 was chosen 13.5% of the times. The observed behaviour implies that the more serious the travelpurpose is the more conservative people tend to be in their choices. Therefore, when the travel purpose changed from meetingwith colleagues to job interview, the preference for route 2 increased along the experience period. In addition, it is possible toobserve that irrespective of the travel purpose, the route of intermediate performance was the least preferred route.

Travellers' route preferences in scenarios 1 and 2 appear to be aligned with findings about route reliability reported inthe literature, i.e., travellers value travel time reliability higher than travel time savings (Small et al., 2005; Liu et al., 2004).Instead of choosing a route that is potentially faster, travellers prefer slower but reliable routes. Figs. 4 and 5 depicttravellers' route choice behaviour in scenarios 1 and 2.

Regarding travellers' reaction and adaptation to information, the behaviour under no information matched our expectations.At first travellers explore the routes to get familiar with their travel times, but, as time evolves and travellers becomemore aware ofthe route characteristics switching rates decrease (Fig. 6). For the sake of clarity, the reader should consider that a route switch is

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observed only if the traveller did not choose the same route as the day before (exploring routes) or as 2 days before (back to usualbehaviour).

When information was provided, three groups regarding compliance rate with information were identified: group 1corresponding to around 45% of the travellers (20 travellers) for which compliance rates with the information were above60%, group 2 corresponding to around 35% of the travellers (15 travellers) for which compliance rates with the information werebetween 30% and 60% and finally group 3 corresponding to about 20% of the travellers (10 travellers) for which compliance rateswith the information were lower than 30%. We hypothesized that the switching rates of travellers that in general do not complywith the advice (group 3) should be similar to the group without information. Fig. 7 shows a tendency of decrease in theexploration rate up to 2/3 of the experiment and then a tendency to increase again. This is not entirely aligned with ourexpectations as it seems that travellers restart exploring the routes. A possible explanation is that although participants in group 3did not tend to comply with the information, they might have felt intrigued about the accuracy of the information and decided toexplore the routes in order to confirm their expectations. As a result a mixed behaviour arose at the end of the experiment, i.e., insome occasions travellers complied with the advice and in others they trusted their own experience.

For groups 1 (high compliance rate) and 2 (medium compliance rate) we argued that as time evolves and travellers becomemore confident about the reliability of the information, switching rates from the information should gradually decrease.Compliance rates with the advice should therefore increase. Our expectations did not resemble the observed behaviour. Averagecompliance rate with the advice for group 1 was around 85%, but very low compliance rates were also observed (Fig. 8). Forinstance, although the lowest compliance rate occurred the day after a very wrong advice, i.e., actual travel times wereunderestimated by more than 15min, lower compliance rates were in general associated to situations in which the informationfavoured route 3 (the least preferred route). The reason why route 3 was the least preferred may be related to the fact that it wasneither the fastest nor the most reliable route. For group 2, no pattern regarding compliance with information was identified as itseems to resemble neither the travel information nor their own experience (Fig. 9). We believe, however, that this may be aconsequence of the design of the experiment inwhich the informationwas neither always correct nor always precise. By correct wemeant that the route indicated to be the fastest was actually the fastest and by precise that the indicated travel times were equal tothe experienced travel times. For instance, at times the information deviated between 10 and 30 min from the true travel times.

Nevertheless, a tendency to follow the information was observed (Fig. 10). In scenario 3, route 1 was the most oftenrecommended, followed by route 3 and finally route 2, and this was the observed pattern for the routes chosen (Fig. 10a).The average compliance rate with the information during thewhole experiment was 63%. In scenario 4, although the informationprovided was almost the same as in scenario 3, compliance rates were around 55%, thus considerably lower. In addition, the

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Scenario 3: Group 2 Scenario 4: Group 2observed pattern for the routes chosen did not fully resemble the pattern of the information provided as route 2 was chosenmore often than route 3 (Fig. 10b).

Similar to the scenarios in which no information was provided, there is a tendency to be more conservative in the choiceswhen the travel purpose is more serious. Moreover, when the information favoured travellers' preferred routes, thecompliance rate was much higher. For instance, as route 3 was the least preferred route, whenever the advice favouredthis route, compliance rates were (relatively) smaller than in other situations (Table 4). The fact that travel times on route 1were usually smaller than on route 3 might have influenced the preference of route 1 over route 3. As a result, when highertravel times on route 1 occurred these were perceived as exceptions. Conversely, higher travel times on route 3 might havebeen perceived as a confirmation that this route was not such a good choice. Therefore, even when the information favoured

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Fig. 10. Travellers' daily route choice in reaction to information in scenarios 3 (Fig 10a) and 4 (Fig 10b). Top part of each figure: the route which the advicefavoured (1, 2 or 3) day by day. Bottom part of each figure: travellers' route choice. Route 1 in black (bottom), route 2 in lilac (middle) and route 3 in darkgrey (top). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

G. de M. Ramos et al. / Journal of Choice Modelling 6 (2013) 173328Table 4Route choice preferences in scenarios 14.

Scenario Travellers' route choice Compliance rate with advice (%)

Route 1 (%) Route 2 (%) Route 3 (%)

1. No info+meeting colleagues 40.5 48.8 10.7

2. No info+job interview 34.5 52.0 13.5

3. Info in minutes+meeting colleaguesOn average 54.7 22.0 23.3 64.0Information favoured route 1 67.1 17.1 15.8 67.1Information favoured route 2 3.6 82.1 14.3 82.1Information favoured route 3 32.0 19.4 48.6 48.6

4. Info in minutes+job interviewOn average 45.4 28.4 26.2 55.0Information favoured route 1 55.5 24.3 20.1 55.5Information favoured route 2 1.1 87.0 12.0 87.0Information favoured route 3 31.3 23.9 44.8 44.8route 3, there was some distrust with respect to its reliability. On the other hand, when the information favoured route 2, itsreliability was accentuated by the fact that route 2 by itself was already reliable. This explains the (very) high compliance rates.

5.2. Role of the reference point in the prediction ability of Prospect Theory

Results regarding the prediction ability of Prospect Theory are aligned with what has been observed in the literature andreinforce the importance of establishing meaningful reference points. The need to align the reference point to travellers'observed behaviour, thus accounting for heterogeneity, can be observed from the data analysis. Discussion regarding therole of the reference point in situations accounting and not accounting for heterogeneity in travellers' behaviour isrespectively presented in Sections 5.2.1 and 5.2.2.

5.2.1. Considering no heterogeneityIn scenarios 1 and 2, in which no information was provided, travellers were more conservative in their choices and

tended to choose route 2 the most. As a result, when the reference point was set equal to the mode of travel times of themost reliable route, the prediction ability of Prospect Theory was the highest. On the other hand, the lowest performancewas observed when the reference point was equal to the average travel time of all routes (Fig. 11a). As under this conditionroute 3 was predicted the most, it is possible to infer that when the reference point is not based on a specific type ofbehaviour, but on average values, routes of intermediate performance benefit from this. In addition, no differences between

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Fig. 11. Role of the reference point under no heterogeneity. (a) Prediction ability of prospect theory in relation to the reference point in scenarios 1 and 2,and (b) Prediction ability of prospect theory in relation to the reference point in scenarios 3 and 4.

setting the reference point as the mode or minimum travel time of the fastest route are observed. Despite the fact that eachof these reference points has its behavioural appeal, as their values were quite similar, differences in travellers' riskybehaviour could not be captured.

In scenarios 3 and 4, in which information was provided, the role of the reference point is more subtle because itsvariability is smaller. Prospect Theory performed best when the reference point was set equal to travel information_2. Thisallows inferring that people tend to rely on actual travel times and use the information solely as a reference of how goodtheir choice is (Fig. 11b). In addition, because route 2 was not often recommended, the worst performance occurred whenthe reference point was equal to the mode of the most reliable route. The results of scenario 4 are somewhat disappointingbecause the prediction ability of Prospect Theory was only slightly better than in scenario 2, in which no information wasprovided, and inferior to that observed in scenario 3, in which only the travel purpose was different. Nevertheless, theseresults were somehow expected because the route choices in scenario 4 were more homogeneous and as the experimentprogressed, route 1 was the only route predicted. For instance, as of the 20th day in 75% of the times the advice favouredroute 1, but travellers chose it only 49% of the times.

The Chi Square test was performed to test the hypothesis that improvements in the prediction ability of Prospect Theoryare due to the values adopted for the reference point. In the scenarios in which no information was provided, we testedthe significance of aligning the reference point to travellers' risk preferences. On the other hand, when information wasprovided, we tested the significance of aligning the reference point to the information provided. These are the nullhypotheses tested.

The Chi Square test was performed taking into account the number of correct predictions of Prospect Theory associated

we cannot reject the hypothesis that aligning the reference point to travellers' behaviour leads to significant improvement in

G. de M. Ramos et al. / Journal of Choice Modelling 6 (2013) 1733 29the prediction ability of Prospect Theory. This suggests that the reference point is aligned with travellers' behaviour. Basedon the outcomes of scenarios 3 and 4 we cannot reject the hypothesis aligning the reference point to the informationprovided significantly improves the prediction ability of prospect theory. This suggests that travellers appear to use theinformation as a reference point.

5.2.2. Accounting for heterogeneityBy considering each traveller as an unique individual, and consequently adopting distinct reference points for each of

them, the prediction ability of Prospect Theory (substantially) increases. This is particularly observed when no informationwas provided (scenarios 1 and 2). Fig. 12 depicts comparisons between the best performance of Prospect Theory under noheterogeneity with the performance of the proposed reference points when accounting for heterogeneity. It is possible toobserve that setting the reference point as defined in rp ii, i.e., the travel times or travel information of the most chosenroute in the 5 days prior to the route choice, definitely leads to major improvements in the prediction ability of Prospect

Table 5Chi Square and significance levels for different reference points under no heterogeneity.

Scenario Reference point Correctpredictions

Wrongpredictions

Chi Squareab ()a

Chi Squarea c ()a

Chi Squaread ()a

1 (a) Aligned with the behaviour of majority 286 314 7.48 112.37 (b) Not aligned with the behaviour of the majority (1) 239 361 (o0.01) (o0.01)(c) Not aligned with behaviour of majority (2) 113 487

2 (a) RP aligned behaviour majority 383 367 (b) RP not aligned with the behaviour of majority (1) 259 491 41.87 171.76(c) Not aligned with the behaviour of majority (2) 141 609 (o0.01) (o0.01)

3 (a) Equal to the information 772 488(b) Aligned with the behaviour of majority 700 560 8.46 8.94 35.64(c) Not aligned with the behaviour of majority (1) 698 562 (o0.01) (o0.01) (o0.01)(d) Not aligned with the behaviour of majority (2) 623 637

4 (a) Equal to the information 733 647(b) Aligned with the behaviour of majority 630 750 15.37 12.26 8.15(c) Not aligned with the behaviour of majority (1) 641 739 (o0.001) (o0.001) (o0.01)(d) Not aligned with the behaviour of majority (2) 658 722

a Significance level.to each of these null hypotheses and some base values. The base values associated to different reference points aredepicted in the third and fourth columns of Table 5. The behavioural appeal of each reference point shown in the secondcolumn of Table 5 reflect (i) the behaviour of the majority, i.e., the route chosen the most in each scenario, (ii) the behaviourof the second most chosen route (Not aligned with the behaviour of the majority (1)), (iii) the behaviour of the routechosen the least (Not aligned with the behaviour of the majority (2)) and (iv) the information provided. Items (i) and (iv)respectively refer to the null hypothesis for the situations in which no information was provided and for the situations inwhich information was provided. It is possible to observe that the Chi Square statistic considering one degree of freedom isvery significant and the significance level is inferior to 0.01 in all cases. As a result, from the outcomes of scenarios 1 and 2

G. de M. Ramos et al. / Journal of Choice Modelling 6 (2013) 1733300%

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Fig. 12. Prediction ability of Prospect Theory with and without accounting for heterogeneity.

Table 6Chi Square and significance levels for different reference points accounting for heterogeneity.

Scenario Reference point Correctpredictions

Wrongpredictions

Chi Squareab ()a

Chi Squarea c ()a

Chi Squareb c ()a

1 (a) Accounting for heterogeneity: rp i 329 271(b) Accounting for heterogeneity: rp ii 398 202 16.61 6.16 42.64(c) Aligned with the behaviour of majority 286 314 (o0.001) (o0.01) (o0.001)

2 (a) Accounting for heterogeneity: rp i 457 293(b) Accounting for heterogeneity: rp ii 486 264 2.40 14.81 29.02(c) Aligned with the behaviour of majority 383 367 (40.10)b (o0.001) (o0.001)

3 (a) Accounting for heterogeneity: rp i 773 487(b) Accounting for heterogeneity: rp ii 841 419 7.96 0.002 8.20(c) Equal to the information 772 488 (o0.01) (40.50)b (o0.01)

4 (a) Accounting for heterogeneity: rp i 765 615(b) Accounting for heterogeneity: rp ii 833 547 6.87 1.49 14.76(c) Equal to the information 733 647 (o0.01) (40.10)b (o0.001)Theory. The real benefit of accounting for heterogeneity may be questioned given that under the reference point as definedin rp i (travel information of the most chosen route up to day d-1), improvements in scenarios 3 and 4 were not sosubstantial. It was observed, however, that despite the fact that on average the number of correct predictions remainedabout the same, an increase in the minimum amount of correct predictions per traveller from 2 to 8 was ascertained. Thiscontributes to the robustness of the model.

The underlying reason for a better performance of the reference point as defined in rp ii rather than as defined in rp i isthat the former allows a quicker adaptation of the reference point to a new route preference and thus to changes inbehaviour. For instance, if a traveller would have chosen route 1 for 10 consecutive days and as of the 11th decided to chooseroute 2, the reference point as defined in rp ii would be aligned with the characteristics of the new route in 3 days, thus onthe 14th day. For the reference point as defined in rp i, however, this would be the case only as of the 21st day when thetraveller had driven on the new route for a period longer than on the previous preferred route.

In addition, the data analysis allows inferring that for travellers who appear to have a clear route choice behaviour ratherthan random route choices, i.e., route switches are observed, but not such that every other day a different route is chosen,accounting for heterogeneity in their behaviour is without doubt the reason of substantial improvements. Thecharacteristics of the routes, however, play an important role and should be very well defined in order to let the referencepoint be able to capture travellers' risky preferences. For travellers that usually preferred route 3, which does not havecharacteristics so well defined, very low prediction rates were observed. This suggests that not only the reference pointshould be aligned with the observed behaviour, but also that the observed behaviour should be a good reflection of theroutes' characteristics.

The Chi Square test tested the hypothesis that improvements in the prediction ability of Prospect Theory are due to thefact that the heterogeneity of the reference point is taken into account (Table 6). This is the null hypothesis tested forreference point values as defined for rp i and rp ii. The behavioural appeal of each reference point shown in the secondcolumn of Table 6 reflect (a) heterogeneity as defined for rp i (null hypothesis), (b) heterogeneity as defined for rp ii (nullhypothesis) and (c) behaviour of the majority, i.e., not taking heterogeneity or information provided into account. Referencepoints aligned with the behaviour of the majority or to the information are the ones that under no heterogeneity led to thehighest prediction ability of Prospect Theory.

It is possible to observe that the Chi Square statistic considering one degree of freedom is very significant and the significancelevel is inferior to 0.01 in the majority of the cases, but not in all of them (Table 6). For instance, we can reject the hypothesis thatimprovements in the prediction ability of Prospect Theory in scenario 2 when considering the reference point as defined in rp ii is

a Significance level.b These values are not statistically significant.

G. de M. Ramos et al. / Journal of Choice Modelling 6 (2013) 1733 31significantly different from considering the reference point as rp i. The same holds for heterogeneity as defined in rp i and theinformation provided in scenarios 3 and 4. These outcomes suggest that other factors, such as travel purpose, might beinfluencing the better performance of Prospect Theory rather than the reference point.

6. Conclusions and future research

The behavioural meaning of the reference point of Prospect Theory and the role of heterogeneity in travellers' routechoice behaviour have been discussed in this paper. This was done by providing a behavioural interpretation to thereference point and, with the support of a case study, comparing the results regarding the prediction ability of ProspectTheory for different reference points and showing the added value of making use of heterogeneous reference points.

The use of Prospect Theory to model traveller behaviour is relatively new. The literature has made clear that there is alack of empirical experiments to validate its use for route choice behaviour, in particular due to the need to betterunderstand the reference point. The presented case study is one of the first attempts to propose a behavioural interpretationto the reference point in a route choice context and to further investigate the role of heterogeneity in the reference point.

6.1. Conclusions

In line with literature, results show that the reference point considerably influences the prediction ability of ProspectTheory, thus reinforcing the need of establishing meaningful values. Moreover, by accounting for heterogeneity in travellers'behaviour, thereby considering that each individual traveller has its own reference point, considerable improvements in theprediction ability of Prospect Theory are observed. The characteristics of the routes also appear to play an important role andresults suggest that not only the reference point should be aligned with traveller's observed behaviour, but also that theobserved behaviour should well reflect the routes' characteristics.

The Chi Square tests performed to test whether improvements in the prediction ability of Prospect Theory were due to aligningthe reference point to some behavioural premises provided significant results. For situations in which no heterogeneity wasconsidered, the null hypotheses tested were that aligning the reference point to travellers' risk preferences and to the informationprovided lead to significant improvements in the prediction ability of Prospect Theory. These null hypotheses respectively refer tothe scenarios in which no information was provided and to the scenarios in which information was provided. When heterogeneitywas taken into account, we tested the null hypothesis that taking into account travellers' heterogeneous behaviour in the referencepoint leads to significant improvements in the prediction ability of Prospect Theory.

The results indicate that we cannot reject the null hypotheses under no heterogeneity, but that one should be carefulwith respect to heterogeneity. For instance, if we are able to easily/quickly identify travellers' preference changes and reflectthese changes in the values of the reference point (for example the switch of preferences from a fast and unpredictable routeto a slower and reliable route), we cannot reject the null hypothesis regarding the importance of taking heterogeneousbehaviour into account. Conversely, heterogeneity in travellers' behaviour does not seem to play an important role.

These outcomes suggest that application of Prospect Theory can highly benefit from the use of meaningful referencepoints that reflect (i) travellers' heterogeneity, (ii) quicker adaptation of preferences over time, and (iii) travellers' (risk)preferences. In addition, it has to be observed that a main drawback of not taking heterogeneity into account is thatirrespective of travellers' preferences, either for reliable and slower or unreliable and faster routes, only the behaviour of themajority can be captured.

Regarding the use of information, results show that despite its potential to influence travellers' behaviour, compliancerates vary a lot among travellers. Nevertheless, travellers' route choices are more clustered when information was provided,in particular when the information was aligned with travellers' preferences for a specific route.

6.2. Implications

The main issue underlying the compliance rate with information is determining the extent to which this impactsnetwork dynamics. For instance, given the high compliance rates in some situations, information provision might influencetravellers to completely switch to alternative routes which would cause a considerable impact in network dynamics.

With respect to the role of heterogeneity in the reference point the results indicate the importance of properly relating travellers'risky-preferences to the reference point. Given the characteristics of our case study, we were able to properly translate the traveltimes of routes 1 and 2 in reference points which were coherent with travellers' observed behaviour/risky-preferences. This,however, was not possible for route 3 due to its mixed characteristics. The consequence of this is that establishing meaningfulreference points for routes withmixed characteristics, such as urban roads, is not straightforward andmakes application of ProspectTheory more difficult.

6.3. Future research

Despite the potential impact of information provision on network dynamics, further conclusions should be drawncarefully. This is due to aspects such as travellers' familiarity with the routes and distances travelled as these are factors that

Gopinath, D.A., 1995. Modeling Heterogeneity in DISCRETE CHOICE Processes: Application to Travel Demand. PhD Thesis. Massachusetts Institute of Technology.Henn, V., Ottomanelli, M., 2006. Handling uncertainty in route choice models: From probabilistic to possibilistic approaches. European Journal of Operational

Research 175, 15261538.

G. de M. Ramos et al. / Journal of Choice Modelling 6 (2013) 173332Hess, S., 2007. Posterior analysis of random taste coefficients in air travel behaviour modelling. Journal of Air Transport Management 13, 203212.Hess, S., A., Stathopoulos, A., Daly, 2011. Allowing for heterogeneous decision rules in discrete choice models: an approach and four case studies. In:

Proceedings of the 2nd International Choice Modelling Conference. July 4th6th, 2011, Leeds, UK.Kahneman, D., Tversky, A., 1979. Prospect Theory: an analysis of decision under risk. Econometrica 47 (2), 263291.Koppelman, F.S., Sethi, V., 2005. Incorporating variance and covariance heterogeneity in the Generalized Nested Logit Model: an application to modeling

long distance travel choice behavior. Transportation Research Part B 39, 825853.Kristensen, H., Grling, T., 1997. Adoption of cognitive reference points in negotiations. Acta Psychologica 97, 27288.Liu, H.X., Recker, W., Chen, A., 2004. Uncovering the contribution of travel time reliability to dynamic route choice using real-time loop data. Transportation

Research Part A 38, 435453.Noland, R.B., Small, K.A., 1995. Travel time uncertainty, departure time choice, and the cost of the morning commute. Transportation Research Record 1493, 150158.Ossen, S., Hoogendoorn, S., 2011. Heterogeneity in car-following behavior: theory and empirics. Transportation Research Part C 19 (2), 182195.Polak, J.W., S., Hess, X., Liu, 2008. Characterizing heterogeneity in attitudes to risk in Expected Utility models of mode and departure time choice.

In: Presented at the 87th Transportation Research Board Annual Meeting. Washington, DC.Prelec, D., 1998. The probability weighting function. Econometrica 66 (3), 497527.Ramos, G.M., Daamen, W., Hoogendoorn, S., 2011. Comparison of Expected Utility Theory, Prospect Theory, and Regret Theory for prediction of route choice

behavior. Transportation Research Record 2230, 1928.Schul, Y., Mayo, R., 2003. Searching for certainty in an uncertain world: the difficulty of giving up the experiential for the rational mode of thinking. Journal

of Behavioral Decision Making 16, 93106.might also influence compliance with information. These were not part of the analysis presented in this paper, but should beobserved in future research.

Due to the important role of the reference point, both its behavioural appeal and the role of heterogeneity should befurther investigated with the support of more empirical experiments. The same holds with respect to the other parametersof Prospect Theory. This will help validating the suitability of Prospect Theory to model travellers' behaviour.

Another aspect to be observed is that although we considered a 10 day experience period in which travellers would beable to explore the routes and get familiar with their characteristics, this paper does not discuss learning. This is because wefocused on investigating the role of heterogeneity in the reference point rather than discussing the role of information onlearning mechanisms. Further analysis regarding the role of learning mechanisms is a topic that should be addressed.

Regarding compliance with information, we argued that non-compliers should behave similar to non-informed travellers.It is not entirely clear, however, why this behaviour was only partially observed (up to 2/3 of the experiment). Futureresearch should investigate reasons for disgruntlement of non-compliers, i.e., if this is due to incorrect advices at thebeginning of the experiment or whatever else might be influencing this type of behaviour.

Acknowledgements

This research is financed by the Dutch Organization of Science (NWO) under the Programme TRISTAM: TravellerResponse and Information Service Technology: Analysis and Modelling. We would like to thank Dr. Enide Bogers for makingavailable the data used in the case study of this paper and the anonymous reviewers for their valuable comments whichhelped improving the quality of our paper.

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G. de M. Ramos et al. / Journal of Choice Modelling 6 (2013) 1733 33

Modelling travellers' heterogeneous route choice behaviour as prospect maximizersIntroductionBehavioural foundations of Prospect TheoryInitial developmentsCumulative Prospect TheoryEvaluating travellers' route choice behaviour as prospect maximizers and the role of the reference pointRole of heterogeneity in the reference point

Case study: investigating travellers' route choice behaviourModel specificationConsidering no heterogeneityAccounting for heterogeneity

Results and discussionTravellers' behaviour in relation to informationRole of the reference point in the prediction ability of Prospect TheoryConsidering no heterogeneityAccounting for heterogeneity

Conclusions and future researchConclusionsImplicationsFuture research

AcknowledgementsReferences