ob
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these theories and models, and illustrates their application using examples of empirical research in travelbehavior research. Arguing that some basic assumptions underlying these theories may not mimic the
Introduction
The analysis and modeling of human ching has a long history in travel behavior re
deringple ches forecute t
vidual and household choice and decision-making processes. Untilthe mid 1970s, spatial interaction and entropy-maximizing mod-els, based on the theory of social physics, dominated the eld(Wilson, 1974; Batty, 1976). Later, random utility theory(McFadden, 1974) and psychological choice theory (Luce, 1959)led to the formulation and application of many discrete choice
oment of choice,le individudge aboutdecision-m
under conditions of certainty.In reality, however, the assumption that attribute valu
certain is not very realistic. When leaving home, individuals willnot be certain about their arrival time to the intended destinationas travel times exhibit inherent uctuation. When choosing publictransport, travelers may not be sure whether they will have a seatas demand and therefore occupancy fundamentally varies on aday-to-day basis. When choosing a place to park their car, driversmay face unexpected queues, and parking garages may be full.
Corresponding author. Tel.: +31 402474527.E-mail address: [email protected] (S. Rasouli).
Travel Behaviour and Society 1 (2014) 7990
Contents lists availab
ou
w.demographics is crucial in evaluating social effects and equity oftransport investment and trafc management scenarios.
Over the last decades, the travel behavior research communityhas applied a variety of theories and modeling approaches to indi-
the attributes of their choice alternatives. At the mthe values of these attributes are invariant, whiassumed to hold perfect and complete knowleattributes. All these models relate to choice andhttp://dx.doi.org/10.1016/j.tbs.2013.12.0012214-367X/ 2014 Hong Kong Society for Transportation Studies Published by Elsevier Ltd. All rights reserved.als aretheseaking
es areroute to arrive at the chosen destinations, etc. Moreover, modelingconsumer response to exogenous policies is essential in assessingthe impact and effectiveness of transportation policies, capturedin terms of changing attributes of the choice alternatives ofinterest. Finally, understanding how choice behavior co-varieswith genders, income categories, age groups and other socio-
popular. In parallel, but less pertinent, researchers have alsoapplied rule-based models to capture decision heuristics (e.g.,Arentze et al., 2000).
Regardless of the modeling approach and the underlying theoryof choice and decision-making, these models have in common theassumption that decision-makers have perfect knowledge aboutshould not come as a surprise, consivel demand forecasting is how peoactivities, decide on the departure tima transport mode, decide where to exquintessence of day-to-day activitytravel behavior, the paper is completed by reecting on the limita-tions of these models and identifying avenues of future research.
2014 Hong Kong Society for Transportation Studies Published by Elsevier Ltd. All rights reserved.
oice and decision-mak-search. This statementthat the essence of tra-oose to participate inthese activities, chooseheir activities, choose a
models (Hensher, 1981; Ben-Akiva and Lerman, 1985). Soon themultinomial logit model became the working horse of the profes-sion, to be complemented by various more advanced, less stringentmodels, such as the nested logit model and generalized extremevalue models, which avoided some of the rigorous assumptionsunderlying the multinomial logit model. Lately, the mixed logitmodel (Train, 2003), MDCV model (Bhat, 2005) and hierarchicalchoice models (Walker and Ben-Akiva, 2002) have become ratherApplications of theories and models of chunder conditions of uncertainty in travel
Soora Rasouli , Harry TimmermansEindhoven University of Technology, Urban Planning Group, Eindhoven, The Netherland
a r t i c l e i n f o
Article history:Available online 27 January 2014
a b s t r a c t
The overwhelming majoriindividuals choose betweecan be questioned as in reimportant therefore that trand predict decision-makinadopt theories and modelstheir applicability to travel
Travel Behavi
journal homepage: wwice and decision-makingehavior research
f models in travel behavior research assume implicitly or explicitly thatlternatives under conditions of certainty. The validity of this assumptiony urban and transportation networks are in a constant state of ux. It isportation researchers develop relevant approaches and models to analyzender conditions of uncertainty. Transportation researchers have tended toiginally developed in social psychology and decision sciences, and exploreice behavior. This invited review paper summarizes the most important of
le at ScienceDirect
r and Society
elsevier .com/locate / tbs
MOHAMMADHighlight
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When choosing a route, drivers can never be sure about the con-gestion situation. Even if the chosen route is normally not con-gested, an accident or sudden adverse weather conditions maycause signicant delays.
Thus, the state of the transportation system and the urbanenvelope are inherently uncertain. Consequently, decision-makersalways face conditions of uncertainty when choosing departuretimes, activities, destinations, transport modes, routes, etc. In thatsense, it is surprising that applications of theories and models of
and management, and suggest some avenues of future research.un f mn;u pn; j 1; :::; J 4
80 S. Rasouli, H. Timmermans / Travel BehaNotation
Most models in travel behavior research on choice and decision-making assume that the probability of choosing a particular choicealternative is some function of the attributes of the choice alterna-tives and a set of socio-demographic variables. The position of thechoice alternatives on these variables is represented by a single va-lue. This represents choice and decision making under conditionsof certainty. In case of choice and decision making under uncer-tainty, the characterization of the choice alternatives is capturedin terms of probability distributions. It implies that individualsare or cannot be sure about the exact state of the choice alternativealong these uncertain dimensions or about the outcome of his deci-sions. This is the realm of choice and decision-making under con-ditions of uncertainty.
To create an overall framework for the various models in an at-tempt to allow the researcher to compare the various approacheswith classic discrete choice model, in this section we will rstintroduce some notation. Expanding the notation suggested byLiu and Polak (2007), assume that each decision maker i is facedwith a set C = {sn; C fSn n Ng of N risky choice alternatives
1 These three theories are just a small subset of theories and models of choicebehavior under uncertainty. For example, Hey and Chris (1997) list the followingadditional theories that were motivated by the inability of expected utility theory toexplain observed behavior. Allais (1952) theory, Anticipated Utility theory, Cumu-lative Prospect theory, Disappointment theory, Disappointment Aversion theory,Implicit Expected (or linear) Utility theory, Implicit Rank Linear Utility theory,Implicit Weighed Utility theory, Lottery Dependent Expected Utility theory, Mach-inas Generalised Expected Utility theory, Perspective theory, Prospective Referencetheory, Quadratic Utility theory, SSB theory, and Yaaris Dual theory. Some of thesetheories are special cases or generalisations of those discussed in this paper; othersdecision making under conditions of uncertainty are relativelyscarce in travel behavior analysis. Moreover, the majority of stud-ies, albeit also small in number, are concerned with uncertainty orvariability in the transportation system (Rasouli and Timmermans,2012a, 2012b), but do not address how individuals make decisionswhen facing such uncertainty and how it affects their activitytravel decisions. Considering the inherent uncertainty in the stateof the transportation system, the formulation and application of(improved) models of decision-making under conditions ofuncertainty should be a eld of research of high priority in travelbehavior research. This paper is meant to provide readers withsome basic background information and a state of the art overview.
In particular, we will discuss the principles and applications ofexpected utility theory, (cumulative) prospect theory and regrettheory as these models have dominated the scarce literature in tra-vel behavior analysis on decision-making under uncertainty.1 Ineach section, we will rst discuss the basic assumptions and speci-cations of the model and some variations, and then discuss selectedresults of empirical studies. In addition to providing a state-of-the-art overview, we will also reect on the appropriateness andlimitations of these models to applications in transportation planningare based on different concepts.2 Although there are differences between risk and uncertainty, we will use the
terms interchangeably in the present paper.i ij i j
Expected utility theory
Principles
Expected utility theory can be traced back to the work of Ber-noulli in 1738 to solve the famous St. Petersburg paradox. This par-adox concerns the problem how much a single gambler should paya casino to enter a game in which he would toss a fair coin, doublesa start gain of 1 unit every time a head appears, and the game endsthe rst time a tail appears. Thus, the gambler would win 2k1
units if the coin is tossed k times before the game ends. The ex-pected payoff of this gamble is equal to E P1k1 12 1 Conse-quently, the expected payoff for the gambler would be an inniteamount of money, implying that the gambler should play the gameat any price. Yet, few people would consider paying any price, giv-ing rise to the paradox: the discrepancy between what peopleseem willing to pay to play the game and the innite expected va-lue. The problem led to a considerable amount of work on decisionmaking under uncertainty. Neumann and Morgenstern (1947)should be viewed as the basis of modern expected utility theoryin that they provided a set of axioms (completeness, transitivityand continuity) to lay the foundation of modern expected utilitytheory.
The most basic version of expected utility theory (EUT), espe-cially used in a normative context, is the expected value model,which states that the overall evaluation or utility uni of prospectsn by decision maker i, given pn can be derived by taking the expec-tation of the outcome evaluations xnij8j over the probability distri-bution pn. That is,(prospects, lotteries).2 Each choice alternative sn in C consists of aset of J possible outcomes or states sn fSnj ;1 j Jg. Each outcomej of the nth risky choice alternative is dened by a vector of observa-ble attributes Xnj fxnjk;1 k Kg. Associated with each risky alter-native is a set of given probabilities pn fpnj ;1 j Jg, such thatPJ
j1pnj 1, where pnj is the probability that outcome Snj is realized
in sn.Choice under risk implies that a decision maker has to integrate
(i) information about the attributes characterizing the risky out-comes, and (ii) information about the probability of each outcome.We assume that each attribute k inuencing outcome j of prospectn is valued according to mapping function h, which translates thevalues of the observable variables xnjk;8j; k into valuation scoresmnijk;8j; k. In turn, the valued variables are integrated according tofunction g to derive to arrive at an valuation mnij;8j of the jth out-come of the nth prospect. Thus,
mnijk hxnjk;8 i; j; k 1
mnij gmnijk;8 i; j 2or,
mnij ghxnjk;8 i; j; k 3Finally, we assume that the overall valuation (or utility) uni of
prospect sn by decision maker i is a function f of the valuationsof the possible outcomes j of the nth prospect, the given probabil-ities of these outcomes pn and a set of model parametersui fuir;1 r Rg that characterize the decision making processin risky situations. Hence,
viour and Society 1 (2014) 7990uni XJj1
pnijxnij 5
Valuation of outcomesSecondly, rather than using the objective outcome values, the
valuation of these values has been used. It follows that
uni XJj1
pnijvnij 12
The prediction of the expected utility model then depends onthe form of h(). Several functional forms have been used in the lit-erature, the popularity of the function often dependent on the do-main of application (cf. Stott, 2006). Table 1 gives an overview.
HARA is in fact a family of models; the CARA, CRRA and DCRA
Behaviour and Society 1 (2014) 7990 81By assuming a decision rule, expected utility can be linked tochoice. The expected utility model assumes that an individual iwillchoose risky alternative sn from choice set C iff
uni > un0i 8sn0sn 2 C 6
When applied in a non-normative context, several lines of researchhave been developed. Note that the prediction in the standard casedepends on (a) given probabilities, (b) outcome values rather thanthe valuation of these values, and (c) deterministic decision rules.Each of these components has been relaxed/replaced.
Subjective probabilitiesIn normative applications of expected utility theory, the proba-
bility of occurrence of a decision outcome is given. In real-life,however, probabilities of outcomes will not be given, but have tobe construed by individuals. It may be assumed that to accountfor this difference, given objective probabilities can be replacedby subjective probabilities or beliefs pni , p
ni fpnij ppnj ;
1 j Jg, where p is some probability weighting function,so that
uni XJj1
pnijxnij 7
For example, Chew and MacCrimmon (1979) suggested a weightingfunction of the following form that gave rise to weighted utilitytheory:
wsnj =XJj0wsnj0 pnj0 8
Sugden (2004) suggested using
wvnj vnj a 9
where a is a risk attitude parameter. Hu and Mehrotra (2012) ar-gued that the BoxCox transformation is promising. It can be ex-pressed as:
wvnj vnj1a if a0
lnvnj if a 0
(10
This transformation was also used in de Lapparent (2010).Quiggin (1982) developed rank-dependent expected utility the-
ory. While the decision weights are independent of the choice setin the previous models, according to this theory, the value of anoutcome depends on the probability of realizing that outcomeand the ranking of the outcome relative to the other possible out-comes. More specically, given an increasing order of outcomes,the decision weight is equal to:
wkj ppnj pnj1 . . . pnJ ppnj1 pnj2 . . . pnJ 11
where p(.) is a nonlinear weighting function with p(0) = 0 andp(1) = 1.
The value ppnj pnj1 . . . pnJ is the subjective weight at-tached to the probability of receiving a payoff at least as good asthe payoff of j, while ppnj pnj1 . . . pnJ is the subjective weightattached to the probability of receiving a payoff strictly better thenthe payoff of j. Obviously the choice of p(.) is critical in specifying arank dependent utility function. If p(.) is concave then p(p) 6 p forall p, which would represent a pessimistic individual, whoover-weights lower ranked outcomes (bad states of the world)
S. Rasouli, H. Timmermans / Traveland under-weights higher ranked outcomes. Vice versa, a convexp(.) represents an optimistic individual, who over-weights thehigher ranked (good states of the world) outcomes.discussed below can be shown to be special cases.Fig. 1 displays the nature of these value functions.
Risk attitudesA major disadvantage of the most basic expected utility model,
the expected value model concerns its insensitivity to the disper-sion of the utilities of the possible outcomes of the choice alterna-tives: decision-makers are assumed to be risk-neutral. To accountfor the effect of dispersion and different risk attitudes, several non-linear transformations of gmnij;ui have been considered, where uis a vector of parameters that affect the shape of g and capturethe decision-makers risk attitudes. If gmnij;ui is concave, the ex-pected utility of a risky prospect will be lower than the expectedvalue, implying risk averse behavior. In contrast, if gmnij;ui is con-vex, the expected utility of a risky prospect will be higher than theexpected value of the prospect, implying that risk-seeking behavioris captured.
Arrow (1965) pointed out the compensation required for a riskaverse individual to accept a gamble. He calculated the probabilityas
p 12 14du00xu0x 13
where d is payoff for the gamble. By dening absolute risk aversionas
a u00xu0x 14
which is a positive constant and integrating Eq. (14), we obtain
ux eax 15Eq. (15) is known as Constant Absolute Risk Aversion (CARA).By dening relative risk aversion which is dependent on ex-
pected value as
b xu00x
u0x 16
integrating, and separating results by b = 1 and b 1, the CRRA(Constant Relative Risk Aversion) utility can be derived:
Table 1Functional forms value function.
Name Equation
Linear m(x) = xLogarithmic m(x) = ln(a + x)Power m(x) = xa
Quadratic m(x) = ax x2Exponential m(x) = 1 exp(ax)Bell m(x) = bx exp(ax)HARA m(x) = (b + x)aLog quadratic m(x) = ln(x+1)+a(ln(x+1)2)
Sumex m(x) = a exp(bx) + cexp(dx)Linear expo m(x) = (ax + b)exp(cx)
Fig. 1. Value function for different parameter values by different methods.
82 S. Rasouli, H. Timmermans / Travel Behaviour and Society 1 (2014) 7990
ux lnx if b 1 17
ever can be interpreted as replacing the assumption ofdeterministic choice with that of probabilistic choice.3 This is
a distribution of possible arrival times, relative to a preferred arri-
S. Rasouli, H. Timmermans / Travel Behaviour and Society 1 (2014) 7990 83accomplished by adding an error term to the utility of the prospect,as commonly assumed in random utility theory.
Consequently, assuming an additive utility function, Eq. (4)then becomes
uni XJj1
f vnij;uipnj eni 23
where eni is an unobservable component of utility associated withrisky choice alternative n for individual i and mnij Znij gnij, whereZnij and gnij are respectively the observable and unobservable compo-nents of the value function associated with outcome j of risky alter-native n for individual i. The probability than individual i will thenchoose risky alternative sn is
pisn Cj PrXJj1
f vnij;uipnj eni >XJj1
f vn0ij ;uipn0
j en0
i
" #8 sn0sn
2 C24
Assumptions about the error terms commonly used in discretechoice modeling can be used to calculate choice probabilities. Mostapplications in travel behavior analysis have used simple logit ormixed logit formulations. However, dependent on theoretical
3 As in random utility theory, alternative interpretations can be given to the errorux x1b
1 b if b1 18
The HARA (Hyperbolic Absolute Risk Aversion) specication ismost exible in that it encompassed both constant and relative riskaversion. By dening risk tolerance as the inverse of risk aversion,we obtain
1a u
0xu00x 19
Assuming linear function ax + b for this tolerance, HARAemerges as a simple hyperbolic function of x:
u00xu0x
1ax b 20
If a = 0, the CARA models is obtained; if b = 0 HARA turns into CRRA.For the general case a 0; b 0 and assuming a > 0 and x > ba
and integrating
ux log x ba
if a 1 21
ux xba
11a
1 1aif a1 22
Probabilistic choiceClassic expected utility theory assumes deterministic choice
behavior. Individual i will choose risky alternative sn from choiceset C iff uni > u
n0i 8 snsn0 2 C. This assumption of deterministic
choice is however unrealistic as individuals may demonstrate dif-ferent choices when faced with the same choice problem. In travelbehavior research, Polak et al. (2008) therefore suggested to com-bine expected utility theory and random utility theory, which how-terms, ranging from an analysts imperfect knowledge and/or measurement of therelevant attributes, individual taste variation, stochastic preferences and determin-istic choice, deterministic preference and probabilistic choice, and even uncertainty.val time. Respondents were shown ten equiprobable positive ornegative delays, relative to the preferred arrival time, xing therange of variation in travel time with reference to the duration ofthe reported trip and the exibility in arrival time at the destina-tion reported by respondents. They compared the performance ofthe expected value model and a constant absolute risk aversionmodel, also allowing for taste variation. Findings indicated thatthe latter type of model outperformed models based on expectedvalue, suggested that risk attitude plays a role in choice of depar-ture time. Particularly, in this study, travelers were found, on aver-age, to be mildly risk averse. However, results evidenced asubstantial degree of heterogeneity in risk aversion.
De Palma et al. (2007, 2012) analyzed the decisions of driverswhether to acquire information and which route to take on a sim-ple congested road network. They considered four information re-gimes: No information, free information, information available fora fee, and private information, which is available free to a singleindividual. Based on their assumption, they showed that privateinformation is individually most valuable, while the benets fromfree and costly information cannot be ranked in general. They alsoderive that free or costly information can decrease the expectedutility of sufciently risk-averse drivers.
Prospect theory
Principles
Kahneman and Tversky (1979) questioned the validity of ex-pected utility theory as a descriptive theory of human choicebehavior. To explain violations of normative expected utility the-ory as reected peoples actual choice behavior, such as Allais par-adox and the empirical nding that people tend to be risk-averseover high probability gains but risk seeking over high probabilitylosses, with magnitude of losses higher than of gains,4 theyformulated an alternative theory of decision making under risk,called prospect theory. According to their original formulation,decision-making processes consist of two stages. In the rst stage the editing stage various decision rules are used to framepossible outcomes in terms of gains and losses, relative to someneutral reference point. Gains refer to outcomes that exceed this ref-erence point, while losses refer to outcomes that fall short. In thesecond stage the evaluation phase the decision maker evaluatesthe outcomes of each alternative according to some value functionassumptions with respect to the nature of the preference function,choice rules and/or distributions of error terms, constant errormodels, probit models, Luce choice models, generalized extremevalue models, etc. can be considered.
Examples
Polak et al. (2008) investigated the inuence of travel time var-iability on mode and departure time choice, using a sample of 215respondents drawn from travelers the LondonBirmingham andLondonReading corridors. Respondents were presented with eightchoice scenarios regarding their trip, which were varied in terms ofuncertainty in their arrival time at the destination due to variabil-ity in travel time. Respondents, who indicated to have a choice ofmode, were requested to make within and between mode compar-isons. Respondents, who did not have any other transport modeavailable, were only asked to make within mode comparisons. Eachalternative was described in terms of a known departure time, tra-vel cost and a distribution of possible travel times. This resulted in4 Other examples are discussed in Avineri and Prashker (2004).
and transforms objective probabilities into subjective probabilitiesaccording to a non-linear probability weighting function. To accountfor violations of classic expected utility theory, the utility functionshould be concave over gains and convex over losses. Empirical nd-ings of risk-seeking versus risk-avoiding behavior require the disutil-ity of a loss to be valued higher than the utility of an equivalent gain.
Let s dene a reference point in the outcome domain. Prospecttheory then states that the utility of prospect n is dened as:
uni XJj1ppnj vnijxnj s 25
Eq. (39) shows that cumulative prospect theory belongs to the
84 S. Rasouli, H. Timmermans / Travel Behaviour and Society 1 (2014) 7990Risky choice alternative sn is preferred to sn0 iff
uni > un0i 8sn
0sn 2 C 26
Tversky and Kahneman (1992) suggested the following functionalform for the value function:
vnijxnj s xnj sa if xnj s 0kjxnj sjb if xnj s < 0
(27
Parameter k > 1 captures the degree of loss aversion, whileparameters a, b < 1 measure the degree of diminishing sensitivityto change in both directions from the reference point. The curveat zero, being steeper for small losses than for small gains impliesrisk averse when outcomes are considered a loss and risk seekingwhen outcomes are considered a gain, implying that people tendto be more sensitive to decreases of their wealth than to increases.
The probability weighting function p is a monotonicallyincreasing function, with discontinuities at 0 and 1, such that itsystematically overweighs small probabilities and underweightslarge probabilities. Several functional forms have been suggestedfor the decision weight function, as shown in Table 2.
Fig. 1 graphs these weighting functions for different values of kand u.
It shows that some of these probability weighting functionshave a reference point, while others do not, and that some func-tions are symmetric, while others are not. In many cases, it willbe impossible to reasons theoretically why one function wouldbe better than any other. Hence, researchers should compare dif-ferent specications. In travel behavior research, applications havetypically applied the functions suggested by Tversky and Kahn-eman, Wu and Gonzalez and Prelec. Fig. 2 graphs these weightingfunctions for different values of k and u
Later, Tversky and Kahneman (1992) extended prospect theoryby including rank-dependent probabilities, allowing for differentprobability weighting for gains and losses (Palma et al., 2008). Inparticular, decision weights in this so-called cumulative prospecttheory are equal to:
ppnj wpnj pnj1 . . .pnJ wpnj1 pnj2 . . .pnJ j 1; . . . ; J 1 and ppnj wpnj 28
Table 2Functional forms for decision weight function.
Goldstein and Einhorn (1987) ppnj kpn
uj
kpnuj
1pnju
Tversky and Kahneman (1992) ppnj pn
uj
pnuj
1pnju 1=u
Prelec (1998) ppnj explnpnj uPrelec II ppnj expklnpnj uWu and Gonzalez ppnj
pnu
j
pnuj
1pnuj
k Luce ppnj expuk 1 pnj kk measures the elevation of the weighting function and u > 0 represents its degreeof curvature.broader class of rank dependent utility models.
Examples
Relatively late, in the early 2000s, transportation researchersstarted to explore the adequacy of (cumulative) prospect theoryto predict traveler behavior under uncertainty. Most applicationsof prospect theory have been concerned with departure timesand route choice behavior under travel time uncertainty.5
In examining the applicability of prospect theory, Jou andKitamura (2002) assumed two reference points: earliest acceptablearrival time and ofcial work start time. Gains and losses were de-ned as non-linear functions around these two reference points,resulting in four-segmented value functions on the earliest arrivaltime, and the work starting time for a given commuter. No empir-ical results were reported. Later, Senbil and Kitamura (2004) addedpreferred arrival time. Senbil and Kitamura (2006) incorporateddelay/early arrival travel time variability directly in the utilityfunction
Schwanen and Ettema (2007) (see also Schwanen, 2007) alsoexamined the applicability of cumulative prospect theory. Theirproblem concerned picking up children at the day care. In particu-lar, they examined the role of reference points in a cumulativeprospect model. Three reference points were used: (i) The timethat most other parents pick up their child; (ii) the time imposedby day care management that the children should be picked up,and (iii) the time that the day care ofcially closes. In the choicetask, only two possible arrival times, rounded to 5 min as muchas possible, were shown. Probabilities were used as commonlyencountered in everyday life and/or rounded to 5% or 10%, and pre-sented in terms of in numbers (1 out of x cases) and percentages.
Some parameters of the cumulative prospect model were arbi-trarily set. A logit specication and a genetic algorithm were usedto calibrate the model. Estimated parameters indicated the valuefunction is slightly convex for early or on time arrivals and concavefor late arrivals, supporting the commonly assumed inverselyS-shaped form. The authors also found evidence of loss aversion,albeit of small magnitude. Finally, they found evidence of over-weighting of small probabilities and underweighting of largerprobabilities, but again the effect was relatively small. The modelbased on cumulative prospect theory tted the data better than amodel, based on expected utility theory.
Xu et al. (2011a,b) contend that the budgeted time to ensurethat a traveler arrives at a certain destination with some desiredprobability serves as the reference point. This time will dependon trip purpose and risk attitude. A low desired probability impliesthat either the trip is not important or the individual is risk seek-ing. The budgeted time reects their subjective beliefs on traveltimes based on their preferences, and can thus potentially serveas reference points (e.g., Lo et al., 2006). Through past experiences,travelers are assumed to develop a reference time for each O-D pairwhich is a function of the budgeted time for each path. In particu-lar, it is assumed that the reference time (point) is the minimum ofthe budgeted times of all paths. Heterogeneity is accounted for bygrouping travelers of a OD pair into different classes with respectto their desired on-time arrival probabilities. A prospect-baseduser equilibrium model where the resulting reference points areconsistent with the equilibrium ow pattern was derived.
Earlier, Connors and Sumalee (2009) also investigated networkequilibrium for different prospect theoretic values of risky linksin a hypothetical network. Based on assumed values of key5 Van de Kaa (2010) offers a considerably more detail overview of relevant studies,not only in the context of route and departure time choice, but also in other domains.Another recent review of the literature is offered by Li and Hensher (2011).
BehaS. Rasouli, H. Timmermans / Travelparameters of prospect theory, they found that a change in refer-ence point values has a signicant inuence on the equilibriumachieved in the network.
Other studies have looked at route choice behavior.Katsikopoulos et al. (2000, 2002) found support for the applicabilityof prospect theory in route choice behavior. Participants in a drivingsimulator received descriptive information about travel time rangesof two routes and were asked to choose a route under differenttrafc scenarios. They found that if travel times on the alternativeroute are on average shorter than those on the reference route,participants demonstrate risk aversion in their route choicebehavior. In contrast, in case of losses, route choice risk-seekingbehavior is observed when the alternative route is riskier relativeto the reference.
Avineri and Prashker (2004, 2005, 2006) also applied prospecttheory in a route choice context. In their rst study, respondents
Fig. 2. Weighted probability functionviour and Society 1 (2014) 7990 85were invited to choose between different routes, characterized bydifferent probabilities of travel times. They found evidence ofnon-linear decision weights and loss aversion. Masiero andHensher (2011) investigated effects of a negative shift of the refer-ence point. A pivoted stated choice experiment framework wasused to estimate the indirect freight transport costs associatedwith a temporary closure of a main road in Switzerland. Logisticsmanagers completed two experiments: the rst designed aroundthe initial reference alternative; the second after road closurearound the second best road alternative. Three pairs of modelswere estimated. The rst pair of models is based on the assumptionof symmetry. The second pair of models is based on referencedependence, estimating different parameters for gains and losses,while the third pair of models further allows for asymmetric non-linearity in gains and losses. Results provided support to the pros-pect theoretic assumption of a change in preference structure, due
for different parameter values.
j1
Behato a shift in reference point. On average, respondents showedevidence of increased loss aversion for cost and time attributes,and a decrease in loss aversion for punctuality.
While these studies focus on stated choice data, Hu et al. (2012)extracted travel time distribution from historical, revealed prefer-ence data. They compared the predictive performance of severalmodels based on expected utility theory and non-expected utilitytheory and concluded that model ts are generally similar withonly prospect theory performing marginally better over expectedutility theory.
Han et al. (2005) applied prospect theory in a typical framingcontext. They assumed that travelers compliance rates with travelinformation does not only depend on the (in)congruence betweenthe information provided and subjective beliefs, but also on thewhether the information is provided in a positive or in a negativesense. Another interesting study on framing was conducted by Fujiiand Kitamura (2004), who postulated that drivers perceive uncer-tain travel time as an interval, and that they frame travel such con-tinuous to compose 5 alternatives of departure time choice interms of the dichotomy of being on time and being late.
The domain of application of original prospect theory was con-cerned with gambling. Consequently, the denition of gains andlosses is clear-cut. The denition of reference points and the cor-responding gains and losses are however not naturally dened inapplications to route, departure and other choices in travelbehavior research. Moreover, reference points likely differ be-tween individuals. Avineri and Bovy (2008) discussed three possi-ble ways to set the value of the reference point in studies onroute choice behavior. A rst option is to use the mean or othersparameters of the travel time distribution, such as the median va-lue or the mode, as a reference point. More specically, Avineriand Prashker (2005) suggested to set the reference point to theactual travel time experienced on average in the population ofthe target traveler group, such as an average of about 30 minone-way commuting time in many countries. In principle, the ap-proach can be further detailed in terms of travel purpose, mode,user group and/or spatial context. Avineri and Prashker (2004) ar-gued that data may be derived from national (or regional) travelsurveys. However, such an approach would deny any regionaland local variability, and will thus likely introduce substantialbias.
A second option is to ask travelers directly about their desiredor ideal travel times. The question here is whether the notion ofgains and losses is conceptually equivalent to the concept of de-sired or ideal travel times. Moreover, respondents may experiencedifculty in expressing what they consider as desired and ideal tra-vel times. Thus, the validity of measurements of desired traveltimes and the value of the reference points remains ambiguous.More methodological research into the measurement of referencepoints and the factors affecting perceived reference points isneeded.
A third option is deriving reference point from stated/revealedpreference data. This approach would involve comparing modelpredictions for different values of the reference point. Further re-search here is required, however, as the parameter space will showmany discontinuities.
A more general problem is that reference points are likely heav-ily context-dependent, and may not only depend on past experi-ences, but also on aspirations and social comparison. Moreover,taste variation may be substantial.
Avineri and Bovy (2008) argued that reference-dependent pref-erences should be based on the perception of reference values.Consequently, they advocated using fuzzy rather than crisp repre-
86 S. Rasouli, H. Timmermans / Travelsentation and extended the concept of reference point to a fuzzyset of reference values, in order to address the vague perceptionof reference travel times for travelers (see also Avineri, 2009).with
Rsnsn0 snj Rsn0 sn sn0
j 30
and R is regret.Note that the reference point in regret theory depends on the
composition of the choice set and the distribution of attribute val-ues across choice alternatives, not on some (context-dependent)arbitrary reference point. It rules out any choice mechanism thatis based on aspiration levels that are not currently attained.
Chorus et al. (2008b) explored the applicability of regret theoryThe empirical studies summarized in this section, indicates thatthe travel behavior literature shows some examples of applicationsof cumulative prospect theoretic models. However, as argued by Liand Hensher (2011), most of these studies have been limited inscope and approach either because (i) they only included particularaspects of prospect theory, (ii) they did not estimate all parametersin a consistent manner, and/or may not have treated the issue of(multiple) reference point in the best possible way. We wouldadd that compared to random utility models, most studies havealso been relatively weak in accounting for stochastic valuation,probabilistic choice and taste variation.
Regret theory
Principles
Seminal regret theory as formulated by David (1982), Fishburn(1982) and Loomes and Sugden (1982, 1987) was originally as amodel of pairwise choice between lotteries (Machina, 1987). It isconcerned with the problem of which of two risky choice alterna-tives, characterized by a set of states that may occur with someprobability and that have a (monetary) outcome, will be chosen.In contrast to expected utility theory and prospect theory, regrettheory is based on the notion that individuals utility or satisfactionof choosing an alternative is not only based on the anticipated pay-off of each individual choice alternative across different states ofthe world, but also on anticipated payoff of the other alternative.More specically, individuals are assumed to consider the possibil-ity that the non-chosen alternative turns out to have a higher pay-off than the chosen one after all. Thus, regret theory incorporatesthe idea that the utility individuals derive from the outcomes oftheir decisions is inuenced by their perception of what wouldhave occurred if they had made different choices. Individuals feelregret when they experience that they would have been betteroff if they had chosen another alternative, and rejoice when theyexperience that they are better off. In some choice situations, theseemotions are responses to ex post evaluations of choices made, rel-ative to foregone alternatives. In other choice contexts, regret isbased on subjective anticipations. Individuals are assumed totrade-off the utility purely derived from the attributes of the choicealternatives against their desire to avoid and minimize future re-gret (and maximize future rejoice). More specically, individualsare assumed to anticipate for each possible state of the choicealternatives the associated regret, dened as the extent to whichthe chosen alternative performs worse than the non-chosen one),and then aggregate these regrets across all possible states of theworld. Thus, regret theory assumes that alternative sn will be cho-sen to alternative sn0 jsn; sn0 2 C iff:
XJpnj Rsnsn0 snj > 0 29
viour and Society 1 (2014) 7990to model travel choices, both under conditions of certainty anduncertainty. Such application needs generalization of the basicmodel from pairwise choice to choice among multiple (risky)
Behachoice alternatives, and from a single attribute to multiple attri-butes. Adopting Quiggins (1994) principle of Irrelevance of State-wise Dominated Alternatives, which states that a choice fromany given choice set is not be affected by adding or removing analternative that is inferior for every state of the world, the regretassociated with any choice, assuming a state of the world to occur,depends only on the actual outcome and the best possible outcomethat could have been attained for that state of the world. It impliesthat for any state of the world, regret associated with a choicealternative only depends on the best available choice alternative.
To generalize to the multi-attribute case, Chorus et al. (2008a)postulated that choice alternatives are valued in terms of the asso-ciated regret on an attribute-by-attribute basis. More specically,regret associated with choice alternative sn, based on a comparisonof a particular attribute with alternative sn0 is equal to zero if sn isvalued equal or better than sn0 on that particular attribute, and anon-decreasing function of the difference in attribute-valuesotherwise. Formally, the linear regret function for attribute k canbe expressed as:
wk max j0;maxfxnjk xn0
jkgj; sn0 2 C 31
and
Rsn Xk
wk 32
Evidently, the linear regret function is a straightforward formula-tion, but may not sufciently capture the magnitude of regret andits relationship with attribute differences. Chorus et al. added aparameter to capture non-linearities in regret. They added thisparameter to the overall regret function. Nonlinear regret, denedat the level of each attribute, would be equal to:
wk max0;maxfxnjk xn0
jk#g; sn0 2 C 33
Compared to linear regret, a concave regret function emphasizesthe importance of small preference differences, whereas a convexregret-function discounts differences. If # =1, across all attributes,regret minimization reduces to utility-maximization.
Later, Chorus (2011) (see also Chorus and Bierlaire, 2013) sug-gested a different non-linear function while in addition it was as-sumed the comparison is not only based on the best choicealternative, but on all foregone alternatives. Thus,
Rns XNn0n
XKk1
ln j1 expbkfxnk xn0
k gj 34
This improved model was motivated by some practical problemsin the estimation of the model. The max operators imply a non-smooth likelihood function, creating difculties in the derivationof marginal effects and elasticities. Moreover, it requires dedicatedsoftware to estimate the model. However, this specication createsother problems. First, the specication is theoretically no longerconsistent with the concept of regret as the value of the regret func-tion is positive also if the chosen alternative turns out to be betterthan all foregone choice alternatives. It can also be easily shownthat Eq. (51) does not approximate zero if the attribute differencesare small and/or the number of competing options is large. Second,if regret would be based on better performance of foregone alterna-tives, the new specication is the same as the old one, except that aconstant is added. The potential problem here maybe that the addi-tion of an arbitrary constant has more effect if attribute differencesare small, and negligible effects if these differences are high. Empir-ically, this function was found to perform better than the linear
S. Rasouli, H. Timmermans / Travelfunction. One might argue, however, that regret is small for smalldifferences, then rapidly increases with increasing attribute differ-ences between the chosen and foregone choice alternative, to tailoroff again with high differences in attribute values. Such preferencewould be captured by a non-symmetric S-shaped curve, as sug-gested in Timmermans et al. (2000). However, this function is moredifcult to estimate.
The comparison against all alternatives rather than the bestforegone alternative is also not necessarily an improvement. It isreasonable to assume that if an individual could have chosen mul-tiple choice alternatives that perform better that the chosen one,regret is higher than if only a single non-chosen alternatives per-forms better, but cognitively this may be quite demanding, partic-ularly in the context of a potentially very large number ofoverlapping routes.
Finally, the assumption of assessment of regret on an attribute-by-attribute basis can be problematic in some contexts. Consider,for example, the problem of destination choice. Imagine, a foregonealternative would have been slightly better in terms of a specicattribute, but it is located substantially farther away than the cho-sen alternative. Would an individual regret his choice; most likelynot, but the assumption of comparison on an attribute-by-attributebasis would imply that the distance attribute would not matter inthis example.
The discussion of regret theory thus far has been based on deci-sion-making under conditions of certainty. For decision-makingunder conditions of uncertainty, additional assumptions need tobe made with respect to the beliefs of the decision-maker aboutthe occurrence of a particular value (or discrete classes) of theattributes for each of the choice alternatives. The most generalassumption would allow for covariances within and betweenchoice alternatives. Let these beliefs be represented by a multidi-mensional probability density function:
Hj Hxnjk;8n; k Yn;k
Hxnjk 35
Then, expected regret is equal to:
ERsn ZjRsnHjdj 36
Individuals are then assumed to choose the risk alternative with theminimum regret. Developments in discrete choice models can beused to derive a probabilistic choice model. For example, realizingthat regret minimization is equivalent to maximizing minus regret,a mixed logit model would imply
Prsn 2 C Zg1 ;d1
expRi;sn giPsn0 2C expRi;sn0 gi
!f gi; didgi; di 37
where di is an individual-specic error term, and gi captures tastevariation.
Very recently, Chorus et al. (2013) suggested a hybrid modelspecication in which some attributes are assumed processed interms of regret, while others in the classic random utility fashion.Although the model was applied to decision making under cer-tainty, it can be extended in a straightforward manner to decisionmaking under uncertainty. Hess et al. (2012) investigated a mix-ture model, allowing some individuals choice behavior to reectregret minimization, and others utility maximization, while alsoallowing for taste heterogeneity.
Examples
Chorus et al. (2008a) compared the performance of regret-theoretical models against utility-maximizing models in thecontext of non-risky choice of transport modes and acquisition of
viour and Society 1 (2014) 7990 87travel information. Using fractional factorial experimental a statedchoice experiments, 252 respondents expressed their choicebetween 31 pairs of car and train, varying trip purposes (business,
ries. Overall, there was not much difference.Beck et al. (2013) compared utility maximizing and regret min-
Behaimizing models for vehicle choice and found regret minimizationto be the preferred behavioral mechanism for groups and individ-uals within groups who show a high degree of responsibility forthe choice of the group. In contrast, there is no difference betweenregret minimization and utility maximization specications forindividuals and group respondents with a lesser role in the ulti-mate group decision.
Reection
This review paper has discussed a selection of theories, modelsand results of choice and decision-making under conditions ofuncertainty. In particular, expected utility theory, (cumulative)prospect theory and regret theory and examples of their applica-tion to mainly route and departure choice decisions were dis-cussed. While some elaborations of expected utility theory,prospect theory and regret theory have in common the goal ofoffering a model that allows the explanation of empirical anoma-lies of choices under conditions of uncertainty implied by the clas-sic normative expected utility theory, they emphasize differentmechanisms and consequently also differ in terms of their mathe-matical specication. For example, while both prospect theory andregret theory use a reference point, in prospect theory this point isexogenous and captures the distinction between gains and losses,whereas the reference point in regret theory is exogenously de-ned in terms of the best foregone choice alternative. In addition,the models differ in terms of the valuing functions and decisionsweight being used.
Each of these theories was originally formulated in disciplinesother than transportation research, and applied to domains otherthan travel behavior analysis. Original theories were formulatedwith gambling in mind. This raises the question about the validityand usefulness of these theories and model in travel behavior anal-ysis. Can aspects of travel behavior be viewed as being congruentcommute, social, leisure), travel times, costs, waiting times andseat availability. Estimation results indicated that regret-basedmodel outperformed the random utility model. Respondentsweighted differences in performance between the chosen andnon-chosen alternative more heavily and tended to discount largerdifferences.
Ramos et al. (2011) used a travel simulator to investigate routepreferences. They asked respondents to make 40 consecutivechoices between three routes in the context of a meeting withco-workers or job interview during the morning commute (depar-ture at 8:00 a.m.; arrival within 1 h). The routes were described asfollows: (i) route 1 consisted mainly of highways, and is the fastestroute, (ii) route consisted mainly of rural roads, and is most reli-able, while route 3 consisted of a mix of highway and urban roads,going through the city centre, and has intermediate performance.Moreover, information provision was varied according to threeconditions: (i) no information provision, (ii) provision of traveltime in minutes and (iii) provision of queue length in kilometers.At the end of each simulated trip, true travel times were provided.Using this data, the authors compared the performance of expectedutility theory, prospect theory and regret theory. They concludedthat the prediction ability of these competing theories varied bytype of information provided. Expected utility theory and regrettheory perform slightly better than cumulative prospect theory ifno information is provided. In contrast, if information is provided,cumulative prospect theory clearly outperformed the other theo-
88 S. Rasouli, H. Timmermans / Travelto gambling?To provide input to a discussion of this question, it should be
realized that all three discussed theories, as they have been appliedin travel behavior research, relate to one-shot decisions and clearlyformulated payoffs. In contrast, route and departure time choicesare never made in isolation. Rather, these decisions are part of dai-ly activitytravel schedules, which often are based on context-dependent, routine-like, habitual behavior. It is not necessarilyclear that the problem is viewed in terms of gains and losses, asprospect theory assumes. Uncertainty is viewed as just a single as-pect of a more complex scheduling problem. Moreover, to the ex-tent uncertainty is important, decision-makers will face multiplesources of uncertainty. A path to a destination will normally con-sist of multiple links and several of those may show high variabilityin travel times, while travelers have the exibility of choosing apath by choosing alternative links between some key nodes inthe network. Uncertainty is not only related to travel times onroutes, but for many activities also on social norms, congestion atthe destination, uncertainty in service levels at the destination,etc. Furthermore, many sources of uncertainty may be interdepen-dent. Decisions about departure under uncertainty will inuencethe uncertainty of travel times, and subsequent uncertainty of ar-rival times and activity duration may impact uncertainty in starttimes and duration of other activities later during the day.
Another major issue that we wish to identify is that decision-making in one shot situations may be inherently different from re-peated choice. In one-shot decision situations, decision-makers donot have any built-up experience about outcome probabilities. Atbest their choices are made on general contentions. Moreover, theycannot alleviate any negative impact of their decisions. In contrast,repeated choices, such as activitytravel decisions, can in principlebe based on accumulated experience. Travelers learn about theoutcome of their decisions and, based on these experienced out-comes, adapt their behavior in reinforcing (the subjective probabil-ity of) positive outcomes and avoiding (the subjective possibilityof) negative or less positive outcomes.
Research agenda
Thus, we argue that most applications of theories and models ofdecision-making under uncertainty in travel behavior researchhave dealt with toy-problems: very simple problems, representedin often too limited hypothetical choice problems, and modeledin ways that do not much justice to context-dependent heteroge-neity and taste differences among travelers. The value of such stud-ies is that they show light and increase our understanding of choicebehavior in the controlled settings. However, if the ultimate aim isto develop comprehensive models of activitytravel behavior ofindividual and household decision-making under uncertainty, thatcan even start to compete with current dynamic models of traveldemand, a tremendous research effort is needed. The good thingabout it is that a research agenda is unfolding easily. Recent at-tempts of improving econometrics underlying models of choiceand decision-making under conditions uncertainty (e.g., Polaket al., 2008; Hu et al., 2012) are an important step forward as it al-lows incorporating probabilistic choices, taste variation, andincomplete knowledge on the part of the researcher. In this con-text, Zhang et al. (2013) proposal to use relative utility theorywhich inherently deals with multiple reference points and canincorporate non-linear functions deserves further attention. Thenext step should be examining more complex decision problems,such as transport mode chains, path choice, combinations of differ-ent choice facets and ultimately complete daily and weekly activ-itytravel schedules, multiple sources of uncertainty, sequentialdecisions under uncertainty, uncertainty in household decision-making and in social networks. Work along these lines in travel
viour and Society 1 (2014) 7990behavior research is still very scarce. Gao et al. (2008, 2010)suggested a routing policy choice model, based on a cumulativeprospect theoretic model, to predict travelers route adaptation to
Behareal-time information and risk attitudes. The approach explicitlycaptures travelers strategic route choice adjustments accordingto information on realized network conditions in stochastic net-works. Sun et al. (2012) considered multiple sources of error andestimated a decision tree model, allowing for latent classes of riskattitudes.
We argue this research agenda is important because basicassumptions underlying the discussed choice theories do not nec-essarily hold under these more complex, dynamic conditions.Barron and Erev (2003) found evidence of risk attitude reversalswhen feedback is introduced in repeated choice experiments.Subjects showed a tendency to avoid risks when faced with lossesand accept more risks when faced with gains. In route-choice,Ben-Elia et al., 2008) obtained similar ndings. They concludedthat travelers mainly exhibit risk-seeking behavior in the shortrun. In the longer run, under reinforcement, they show a ten-dency to avoid risk. They also did not nd any evidence of lossaversion and the prospect theoretic model specication was notbetter in terms of model t compared to an expected utility spec-ication. Before, Erev et al. (2008) also questioned the validity ofprospect theory in explaining decision-making in repetitive situ-ations, and argued that it is better to be described in terms ofdiminishing sensitivity to absolute payoffs. Avineri and Prashker(2005) found that the higher the variance in travel times is, thelower travelers sensitivity to travel time differences. This behav-ior leads to patterns, which differ signicantly from those predic-tions of cumulative prospect theory.
Learning also plays a case role in regret theory (Ben-Elia et al.,2012). In order to judge the difference in utility between a chosenalternative and foregone alternatives, decision-makers need toexperience and learn the probability of outcomes of alternativechoices. It creates however an interesting problem. Traveler willexperience and learn the state of the world for the chosen alterna-tives, but not for the foregone alternatives, unless they actively col-lect information and that information is available.
All theories considered in this review paper are based on thenotion that individuals will maximize some individual parameter-ized (subjective) (reference-dependent)-utility or value function.However, experienced travel times are the aggregate result ofaccumulated individual decisions. It means that the expected tra-vel time at any moment of choice does not only depend on thedecision of an individual, but also on the aggregated results ofthe decisions of all other individuals. Thus, in a fundamental sense,expectations should take the expected decision of all others intoaccount. Rational behavior of any individual may include strategicbehavior in the sense it is based on the anticipated decision ofother individuals. The situation is however more complex in thesense that the other individuals may also act strategically, leadingto an iterative loop. Under normal circumstances, the described is-sue may just be random noise. However, when travel recommen-dation is provided, its importance may be quite different. Hanet al. (2008) is the only work along these lines in route choicebehavior. They found that individuals were able to anticipate thestrategic behavior of other travelers under uncertainty and adjusttheir own choice accordingly. The topic of strategic behavior underconditions of uncertainty can be more systematically included inthe various approaches discussed in this paper.
In summary, the need to focus research in travel behavior onchoice behavior under uncertainty cannot only be argued from atheoretical point of view, but is also supported by results of a lim-ited number of empirical studies, which lead to the unavoidableconclusion that observed dynamic patterns are not congruent withpredictions of commonly used theories. Further development and
S. Rasouli, H. Timmermans / Travelthe formulation and exploration of alternative, more encompassingtheories and models are required. We hope that this invited reviewarticle may be a trigger and inspiration for scholars to address theidentied issues and avenues of future research.
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Applications of theories and models of choice and decision-making under conditions of uncertainty in travel behavior researchIntroductionNotationExpected utility theoryPrinciplesSubjective probabilitiesValuation of outcomesRisk attitudesProbabilistic choice
Examples
Prospect theoryPrinciplesExamples
Regret theoryPrinciplesExamples
ReflectionResearch agendaReferences
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