Spatial choice: a matter of utility or regret? - Semantic Scholar

Environment and Planning B: Planning and Design 2009, volume 36, pages 538 ^ 551

doi:10.1068/b34092

Spatial choice: a matter of utility or regret?

Caspar G Chorus

Section of Transport Policy and Logistics Organisation, Delft University of Technology, PO Box 2628, Delft, The Netherlands; e-mail: [email protected]

Theo A Arentze, Harry J P Timmermans

Urban Planning Group, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands; e-mail: [email protected], [email protected] Received 31 August 2007; in revised form 14 January 2008; published online 26 January 2009

Abstract. This paper provides an empirical comparison between utility-maximization and regretminimization perspectives of spatial-choice behaviour. The key difference between these two perspectives is that the regret-minimization perspective implies that the anticipated satisfaction associated with a chosen spatial alternative depends on the anticipated performance of nonchosen alternatives. In order to provide a meaningful statistical comparison, we formulate a model of regret minimization such that it reduces to utility maximization for a given parameter restriction. Estimation results, based on a binary stated travel-mode-choice experiment, show how the regret-based model outperforms its utilitarian counterpart. Furthermore it is shown how participants in the experiment attached relatively much weight to the situation where the nonchosen alternative is slightly better than the chosen one, and they tend to discount larger differences. We show how this concavity of the regret function is in line with the prospect theoretical notion of risk-seeking behaviour in the domain of losses.

1 Introduction It is widely acknowledged that understanding people's spatial-choice behaviour is critical for the successful planning and design of the spatial structure of cities and regions. As a result, past decades have seen much attention being devoted to the development of descriptive spatial-choice models (see, for example, Desbarats, 1983; Leonardit and Papageourgiou, 1992; Maat and de Vries, 2006; Nijkamp et al, 1997; Pirie, 1976; Thill and Wheeler, 2000). For many years now, the majority of these spatial-choice models have been based on utility-maximization premises. As an illustration, many examples of utility-maximization-based spatial-choice models can be found in the context of residential choice (for example, Bhat and Guo, 2004), housing choice (for example, Molin et al, 1996), destination choice (for example, Limanond and Niemeier, 2003), travel-mode choice (for example, Hensher and Rose, 2007), telecommuting (for example, Mokhtarian and Salomon, 1994), and route choice (for example, Frejinger and Bierlaire, 2007). Notwithstanding the success of the utility-maximization paradigm in the context of spatial-choice modelling, a large number of alternative behavioural perspectives have also been proposed through the years. Regret minimization (Chorus et al, 2008) is a behavioural perspective that has been introduced recently in the context of spatial-choice modelling as an alternative to utility maximization that has been popular for many years in such areas as microeconomics (for example, Hart, 2005), psychology (for example, Crawford et al, 2002), and marketing (for example, Inman et al, 1997). Regret can be anecdotally described as what you experience when standing in line at a cash register and observe that while your line makes no progress an adjacent line does. The fact that another line performs better than the one you chose makes the situation worse than the situation where all lines are moving as slowly as yours.

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C G Chorus, T A Arentze, H J P Timmermans

Taking this observation one step further, decision-making literature suggests that one is likely to anticipate this regret when choosing which line to join (Zeelenberg, 1999). Following comparable lines of argumentation, a large number of studies in various areas of the decision sciences have applied the notion of (anticipated) regret minimization as a determinant of behaviour choice (for example, Engelbrecht-Wiggans, 1989; Lemon et al, 2002; Loomes and Sugden, 1982; Simonson, 1992; Zeelenberg and Pieters, 2007). Regret minimization clearly contrasts with utility maximization as a descriptive model of choice. It assumes that the anticipated level of satisfaction associated with choosing an alternative depends on the performance of other foregone alternatives, whereas utility maximization postulates that anticipated satisfaction depends only on the performance of the chosen alternative.(1) Inspired by this fundamental contrast between the workhorse of spatial-choice modelling, utility maximization, and the emergent perspective of regret minimization, this paper provides an empirical comparison between the performance of these two approaches. More specifically, we adopt a discrete-choice perspective on spatial-choice behaviour, where the inability of the analyst to observe faultlessly a decision maker's utility (regret) is modelled by random error components. The discrete-choice paradigm has been, for more than two decades, the standard model of econometric analysis of utility-maximizing spatial-choice behaviour (see, for example, Ben-Akiva and Lerman, 1985; Dugundji and Walker, 2005; Pinkse and Slade, 1998). Furthermore, the discretechoice framework has been shown to be compatible with the notion of (random) regret minimization as well (Chorus et al, 2008). As such, it provides a suitable platform for empirically comparing utilitarian and regret-based approaches. To the best of our knowledge, an emprical comparison between discrete-choice models based on the premises of utility maximization and regret minimization has not been made in a spatial-choice context. Moreover, it appears that, notwithstanding the interest in the regret-based approach in other areas of the social sciences, empirical comparisons with its utilitarian counterpart are virtually nonexistent. Hey and Orme (1994), examining choices between lotteries, forms an exception. We take the following approach: first, we consider travel-mode choices in our empirical analyses. However, we feel that our results will be, to a reasonable extent, applicable to other spatial-choice contexts, for example involving residential destination or route choices. Second, we consider binary choices. The reason for this is that we were able to develop for the binary situation a regret-based model that reduces to a utility-based model given one particular parameter restriction. This provides the opportunity to apply statistical testing for nested hypotheses when comparing the two models. Finally, in both our model and the dataset there is no uncertainty attached to the attributes of the alternatives. Although most regret-based models focus on choice under uncertainty (or risky choice), there is no reason to ignore the role of regret in choices made under conditions of full certainty. The key ingredient of the regretbased approach, in our opinion, is that the anticipated level of satisfaction associated with an alternative depends on the anticipated performance of nonchosen options as well. As will become clear in the next section, our regret-based model captures this notion and as such contrasts with its utilitarian counterparts. Finally, note how this paper contributes to another paper written by the same authors, that recently introduced a regret-based travel-choice model (Chorus et al, 2008). (1) Note

that we here refer to the standard neoclassical notion of utility maximization (Von Neumann and Morgenstern, 1947). Other forms of utility maximization, such as the relative utility concept (Zhang et al, 2004), have suggested ways to incorporate the performance of nonchosen alternatives in the utility function of a considered alternative.

Spatial choice: a matter of utility or regret?

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First, whereas the earlier paper introduced random regret-minimization models as a possible alternative to random utility-maximization models, this paper provides an explicit empirical comparison between the paradigms. Second, this paper extends the model proposed in the earlier paper by specifying a concavity ^ convexity parameter. Not only does this parameter enable a meaningful statistical comparison between the two paradigms, it also has an elegant behavioural interpretation that we discuss in depth in this paper. The analyses presented in this paper are based on a different dataset than the one discussed in the earlier paper. Thus, the current paper presents an application of an extended version of the model presented in Chorus (2008), and provides an empirical comparison between the regret-minimization paradigm and random utility-maximization paradigm using a different dataset. In section 2, we construct a regret-based model of (binary) spatial choice that reduces to a utility-based model for a particular parameter restriction. We also show how the regret-based model may lead to different choice outcomes than would be obtained through utility maximization. Section 3 proceeds by introducing the binary stated mode-choice dataset that is used for empirical analysis. Section 4 presents estimation results and statistically compares the performance of the utility-based and regret-based models. In addition, we discuss how our results are in line with the prospect-theoretical notion of risk seeking in the domain of losses (Kahneman and Tversky, 1979). Section 5 presents conclusions and avenues for further research. 2 Regret-based and utility-based binary spatial-choice models We first propose a regret-based and a utility-based model of binary spatial choice, and subsequently show under which restriction the regret-based model reduces to utility maximization. 2.1 Regret-based model

We derive our binary regret model from the multinomial random regret-minimization conceptualizations proposed by Chorus et al (2008). Consider a traveller who faces a choice between alternatives i and j. The alternatives are fully defined in terms of the attributes x and y and a dummy variable z: i ˆ fxi , yi , zi g and j ˆ fxj , yj , zj g. One may, in the context of travel-mode choice, envisage x and y representing travel times and travel costs, respectively, of a car option and a train option. Variable z then indicates whether a particular mode alternative is car travel or not. We assume that travellers evaluate the two alternatives in terms of the associated regret on an attributeby-attribute basis. Furthermore, we hypothesize that the regret that is associated with alternative i due to a comparison with alternative j based on a particular attribute equals zero in the case when alternative i scores equal or better than j on the particular attribute. Otherwise, the regret associated with the attribute comparison is a nondecreasing function of the difference in attribute values. Here we define this attribute regret in terms of linear functions jx , jy , and jz , where bx , by , and bz reflect the traveller's perceived relative importance of the different attributes. Note that the analyst does not prespecify whether high or low attribute values are `better'. Instead, this is something that emerges from the estimation process (specifically, the signs of the estimated parameters). jx …xi , xj † ˆ maxf0, bx …xj ÿ xi †g , jy … yi , yj † ˆ maxf0, by … yj ÿ yi †g , jz …zi , zj † ˆ maxf0, bz …zj ÿ zi †g .

(1)

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C G Chorus, T A Arentze, H J P Timmermans

We proceed by conceptualizing the level of overall regret associated with an alternative as a function of the sum of the regret associated with the attribute-by-attribute comparisons with the other available alternative. This summed attribute-by-attribute regret can be considered to be a measure of the degree in which the nonchosen alternative performs better than the chosen one, on attributes where i is not the preferred alternative. It incorporates the traveller's preferences b. Adding an independent and identically distributed extreme value error component e (with variance p2 =6) gives a random regret (RR) formulation. This error term reflects the analyst's measurement errors in combination with his failure to capture all attributes that are relevant to the decision maker, as well as the mistakes and idiosyncrasies that travellers may display when making a choice.  # RRi ˆ jx …xi , xj † ‡ jy …yi , yj † ‡ jz …zi , zj † ‡ei , (2)  # RRj ˆ jx …xj , xi † ‡ jy …yj , yi † ‡ jz …zj , zi † ‡ej . The alternative with minimal overall regret is chosen. Parameters b are to be estimated from observed choices. Parameter # can be regarded as a concavity ^ convexity parameter, transforming the summed attribute-by-attribute regret into overall regret.(2) If # > 1 (implying convexity), this means that the individual puts extra weight on high levels of summed attribute-by-attribute regret, and tends to ignore relatively low levels. If # < 1 (implying concavity), the individual attaches extra weight to relatively low levels of attribute-by-attribute regret, while discounting relatively high levels. Thus, the random regret-minimizing traveller chooses mode i if and only if:  # maxf0, bx …xi ÿ xj †g ‡ maxf0, by …yi ÿ yj †g ‡ maxf0, bz …zi ÿ zj †g  # ÿ maxf0, bx …xj ÿ xi †g ‡ maxf0, by …yj ÿ yi †g ‡ maxf0, bz …zj ÿ zi †g > e  , (3) with e being the (logistically distributed) difference in random errors. Acknowledging that minimizing regret is equal, in a mathematical sense, to maximizing negative regret, it is directly seen that equation (3) implies that choice probabilities for modes i and j are given by straightforward binary logit functions: P…i† ˆ

exp…ÿRi † exp…ÿRi † ‡ exp…ÿRj †

and

P… j† ˆ

exp…ÿRj † , exp…ÿRj † ‡ exp…ÿRi †

(4)

where Ri ˆ RRi ÿ ei and Rj ˆ RRj ÿ ej . 2.2 Utility-based model

Consider exactly the same situation as above, and now consider a utility-maximization traveller. We assume the following straightforward additive linear random utility functions: Ui ˆ bx xi ‡ by yi ‡ bz zi ‡ ei , Uj ˆ bx xj ‡ by yj ‡ bz zj ‡ ej .

(5)

Here, the random errors are also independent and identically distributed extreme values (with variance p2 =6). As a consequence, the random utility-maximizing traveller chooses mode i if and only if: bx …xi ÿ xj † ‡ by …yi ÿ yj † ‡ bz …zi ÿ zj † > e , (2) As

(6)

such, # is the counterpart of the Q-function that plays a central role in the original microeconomic regret-theoretical framework developed by Loomes and Sugden (1982, page 810). Note however, that our model differs from this original framework in the sense that we consider multiattribute choice and that we do not hypothesize a symmetrical regret ^ rejoice function.

Spatial choice: a matter of utility or regret?

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where e being the (logistically distributed) difference in random errors. This implies that choice probabilities are also given by straightforward binary logit functions: P…i† ˆ

exp…Vi † , exp…Vi † ‡ exp…Vj †

and

P… j† ˆ

exp…Vj † , exp…Vj † ‡ exp…Vi †

(7)

where Vi ˆ Ui ÿ ei and Vj ˆ Uj ÿ ej . 2.3 How and when regret minimization differs from utility maximization

Consider the situation where # ˆ 1. This is the case when decision makers do not put extra weights on either small or large performance differences between the two alternatives. Then, overall regret equals summed attribute-by-attribute regret. Consider two travellers: one maximizes utility, the other minimizes regret with # ˆ 1. Both travellers face the same binary mode-choice situation, being the one described above, and have the same set of preferences bx , by , bz . Setting # ˆ 1 in equation (3), it follows directly that the condition stated in equation (3) equals that stated in equation (6). As a result, regret minimization (as conceptualized above) reduces to utility maximization when # ˆ 1.(3) Note that, in the context of multinomial choices, regret minimization only reduces to utility maximization when # ˆ 1 and there exists an alternative which performs as well as or better than all other alternatives, in terms of every single attribute [see Chorus et al (2008) for a proof]. We now illustrate how # 6ˆ 1 may lead to different choice outcomes between the two paradigms. Consider the following arbitrary attribute values for modes i and j (where x may be conceived as travel time, y as travel costs, and z as a dummy variable for a car option): i ˆ fxi ˆ 50, yi ˆ 4, zi ˆ 1g, j ˆ fxj ˆ 65, yj ˆ 3, zj ˆ 1g. Assume bx ˆ ÿ0:1, by ˆ ÿ1, bz ˆ 1. A utility maximizer chooses mode i if and only if [filling in the attribute values and preferences in equation (6)]: e < ÿ0:5. A regret minimizer chooses i if and only if [filling in the attribute values and preferences in equation (3)]: e  < 1:5# ÿ 2# . As was discussed above, this latter condition is equivalent to the condition given by the utility-maximization approach for # ˆ 1. However, for # 6ˆ 1, the condition posed by the regret model differs. For example: if # ˆ 2, implying a convex regret function, the regret condition becomes: e  < 2:25 ÿ 4, or e < ÿ1:75. For those values of e  that satisfy both conditions (that is, for e  < 1:75) or satisfy none of them (that is, for e  > ÿ0:5), regret minimizers and utility maximizers arrive at the same choice. However, for ÿ1:75 < e  < ÿ0:5, utility maximizers would choose mode i while regret minimizers would choose mode j. A similar illustration can be given for # < 1, implying a concave regret function. Take for example # ˆ 0:5, then the regret condition becomes: e  < 1:22 ÿ 1:41, or e  < ÿ0:19. Again, for those values of e  that satisfy both conditions (that is, for e  < ÿ0:5) or satisfy none of them (that is, for e  > ÿ0:19), regret minimizers and utility maximizers arrive at the same choice. However, for ÿ0:5 < e  < ÿ0:19, utility maximizers would choose mode j while regret minimizers would choose mode i. These differences in choice conditions between the different forms of the overall regret function (linear versus convex versus concave) result in different choice probabilities for different functional forms of overall regret. Figure 1 visualizes this by showing choice probabilities for the train option described in the following binarychoice situation: train ˆ fxi ˆ 50, yi ˆ 4, zi ˆ 0g, car ˆ fxj ˆ 50, yj ˆ 4, zj ˆ 1g. In other words, car and train only differ in terms of unobserved attributes, captured in a car-constant bz . We vary this car constant between ÿ3 and ‡3 and compute, using (3) Readers

familiar with microeconomic regret theory may note the analogy of this result with the fact that a linear Q-function makes the original, single-attribute symmetrical version of regret theory reduce to expected-utility maximization (Loomes and Sugden, 1982).

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P (train) percentage

120 Linear regret (ˆ utility max)

100 80

Concave regret

60 Convex regret

40 20

2:6

3:0

2:2

1:4

1:8

0:6

1:0

0:2

ÿ0:2

ÿ0:6

ÿ1:0

ÿ1:4

ÿ1:8

ÿ2:2

ÿ2:6

ÿ3:0

0 Car constant

Figure 1. The effect of functional form of regret function on choice probabilities for car and train options.

equation (3), the probability that train is chosen for different values of #. # ˆ 1 is represented by the curve `linear regret (ˆ utility max )', # ˆ 2 by `convex regret' and # ˆ 0:5 by `concave regret'. All three curves show that a larger preference for the car option results in a lower choice probability for the train option. However, it is directly seen that differences in choice probabilities do arise for different specifications of the functional form of the overall regret function. Around the point of choice indifference [car constant ˆ 0; P(train) ˆ 50%], it appears that, when compared with the well-known linear-in-parameters logit function for linear regret, choice probabilities are less (more) sensitive to changes in the car constant for convex (concave) regret. A concave regret function stresses the importance of small differences in preference for car or train, whereas convex regret discounts these small differences. Conversely, the concave regret function becomes less sensitive to changes in car preference when preferences are more pronounced. As a result, the concave regret function still presents a nonnegligible choice probability for the train option in cases where a strong preference for car exists and vice versa, whereas convex regret gives zero choice probability for the car option for stronger preferences for the train option. When car constant equals 1, the three functions coincide. This is expected, since 1# ˆ 1 for all values of #. These differences in choice probability functions make it possible to estimate the value of # from observed choice data, and infer whether the data-generating process was driven by (convex or concave) regret minimization or by utility maximization (linear regret minimization). 3 Data collection Participants to our binary mode-choice experiment were recruited through placement of advertisements in a campus newspaper and a free newspaper. Also an e-mail was sent to about 500 students. Criterion for participation was that participants had some experience with travelling by both car and train. A 20 reward was offered for participation. In total, 261 individuals were recruited this way. For 252 of them, complete choice records were obtained. Table 1 presents some response group characteristics of these 252 participants. On the basis of these characteristics, it can be said that as far as sociodemographics (gender, age, education level, main out-of-home activity) are concerned, the sample has reached a rather high level of heterogeneity. It should be noted here specifically, that the majority of participants (about 55%) were not affiliated to our university as either students or academic staff. However, it is clear that our sample is not representative of the average traveller in terms of travel-mode-choice behaviour;

Spatial choice: a matter of utility or regret?

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Table 1. Response group characteristics (n ˆ 252) Variable

Frequency (%)

Gender Female Male

48 52

Age (years) < 25 < 40 5 40

50 23 27

Completed education Lower education 1 Secondary school 66 Higher education 33 Main out-of-home activity Paid work 43 Education 45 Other 12 Table 2. Trip purposes. Context

Text

Business

Please envisage the following : ``You are on your way to an important business meeting, which starts at exactly one hour from now. '' You may also envisage the following : ``Your are on your way towards a job interview for an internship as part of your education. ''

Commute

Please envisage the following : ``You are on your way to work. Your working day starts within an hour from now. You have no appointments today. '' You may also envisage the following : ``You are on your way to the university library, where you want to study. ''

Social

Please envisage the following : ``During your leisure time, you are on your way for a lunch with friends at their home. You told them that you would be there in about an hour. ''

Leisure

Please envisage the following : ``During your leisure time, your are on your way towards a museum, on your own. The day is still young; you have plenty of time to visit the museum. '' You may also envisage the following : ``You are on your way to the beach or the woods, for a walk''.

in particular, participants are likely to be less captive with respect to either car or train than the average traveller, which is a direct result of the above mentioned criterion for participation. For the present purpose of comparing utility-based and regret-based models, this does not seem to be a problematic issue. Notwithstanding that we feel that this bias is likely to have no serious implications for the current study, it may be kept in mind when considering the analyses and conclusions presented in this paper. Each session followed the same programme. After an introduction, a web survey concerning the participant's actual travel behaviour was completed. Subsequently, the stated travel-mode-choice experiment was performed. This experiment consisted of a series of binary car ^ train mode choices. Trip purposes were randomly assigned (see table 2).

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Table 3. Attribute level variation for choices of car or train travel. Attribute

Travel time (min) Travel cost ( ) Waiting time (min) Seat availability

Attribute level car

train

40/50/60/70 0/3.5/7.0/10.5 ± ±

30/40/50/60 0/3.5/7.0/10.5 5/15 during entire trip/not during first 15 min

Car and train alternatives were specified in terms of their travel times and costs, train alternatives additionally in terms of the waiting times and seat availability. Attribute values were varied systematically across alternatives (see table 3). A fractional factorial experimental design was applied that resulted in 16 binarychoice tasks per individual. Participants did not have the possibility of rejecting both alternatives. In total, 252  16 ˆ 4032 stated mode choices were used for the present analysis, of which 2282 were choices for car (57%). 4 Regret versus utility: an empirical test To empirically test the performance of the utility-maximization and regret-minimization approaches, we estimate the parameters of the random regret model as given by equation (2) on our binary mode-choice dataset. We added together waiting and travel times of train alternatives, giving total travel time. One generic parameter is estimated for travel times by either car or train. After a preliminary analysis, it was decided to dummy code trip purpose `business' for the `important business meeting' context and have it interact with travel time. Seat availability entered the model as an addition to a train option's constant. Should the #-parameter presented in equation (2) turn out to be (statistically) indistinguishable from 1, this implies that the observed data result from utilitymaximization-choice behaviour or, equivalently, from regret-minimization-choice behaviour with linear regret function. Should the estimate for # be less than 1, this implies that the data is generated by a process of regret minimization and that the regret function is concave. In other words: much weight is put on small differences in performance between the two alternatives, whereas larger differences tend to be discounted. By analogy, # > 1 signals that the regret-minimization paradigm outperforms utility maximization, and that the regret function is convex. Small differences in performance between the two alternatives tend to be ignored, whereas larger ones are given extra emphasis. We coded the regret-based model presented in section 2.1 in GAUSS 7.0, applying the MaxLik module for maximum-likelihood estimation.(4) Table 4 shows the estimation results of the unrestricted regret-based model and the regret-based model with # ˆ 1 (shown in section 2 to be equivalent to a utility-based model). (4) Note that the panel structure of the dataset (each participant made sixteen choices) would have allowed for the estimation of one or more random-agent effects. However, it appeared that difficulties occurred while inverting the Hessian when we tried to estimate such an agent effect (an individual-specific preference for car over train) together with parameter #: for some numbers of intelligent draws used for the evaluation of the agent-effect integral, the Hessian did invert, while for others, it did not. In those cases where the Hessian did invert, correlations between estimated parameters where rather high. Together, this signals that our dataset does not support simultaneous estimation of the agent effect together with parameter #. Therefore, we chose to incorporate the panel structure indirectly, by using robust t-values.

Spatial choice: a matter of utility or regret?

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Table 4. Estimation results for the unrestricted regret-based model and for the regret-based model with # ˆ 1(a utility-based model).

Car constant Travel time Travel costs ( ) Travel time  business trip a Seat availability (on train) Concavity ± convexity parameter (#)

Regret-based (# estimated)

utility based (# ˆ 1)

parameter

robust t-value (wrt 1)

parameter

robust t-value

1.491 ÿ0.083 ÿ0.501 ÿ0.116 1.324 0.654

5.763 ÿ7.929 ÿ7.020 ÿ4.273 6.633 16.720 (ÿ8.849)

0.839 ÿ0.049 ÿ0.259 ÿ0.051 0.734 1

9.174 ÿ17.213 ÿ20.082 ÿ5.408 8.581 ±

0-log-likelihood log-likelihood at convergence Adjusted r 2 (number of parameters) AIC b (lower AIC ! better model) BIC c (lower BIC ! better model) Number of cases

ÿ2794.8 ÿ1756.8 0.369 (6) 3525.6 3563.4 4032

ÿ2794.8 ÿ1780 0.361 (5) 3570.0 3601.5 4032

a To

be added to the travel time parameter. ˆ Akaike information criterion. c BIC ˆ Bayesian information criterion. b AIC

Although it is not the aim of this paper to identify the importance of modal attributes, we will briefly consider the estimated b-parameters. All estimated traveltime and cost parameters are of the expected sign, and imply reasonable values of time, respectively 10 (11) per hour for nonbusiness trips, 24 (23) per hour for business trips for the unrestricted regret-based model (restricted regret-based model/ utility-based model). Note the similarity between the parameter ratios implied by the regret-based and utility-based models. Both models show a positive and significant average preference for car over train, and a positive and significant valuation of seat availability when travelling by train. Note, however, that b-parameters, estimated within our regret-based framework, have a different meaning from those estimated within a utilitarian framework. Regret bs represent the potential contribution of an attribute to the regret associated with an alternative. An attribute's actual contribution to regret depends on whether the considered alternative performs better or worse on the attribute than the alternative it is compared with [see equation (1)]. As a result, the implied value of time at 10 per hour should be interpreted as follows in a regret-based context: a cost difference between alternatives of 1 potentially adds, for the average traveller, about six times as much regret to the considered alternative as does a time difference of 1 min. This brings us to discussing the estimate for #, which, as was shown above, provides us with information about the nature of the travellers' choice processes. A high t-value (16.72) shows that our estimate for # (0.654) differs from zero in a statistical sense. However, for this present paper it is of much more importance whether # differs significantly from 1: a t-value of ÿ8:85 shows that it does. Furthermore, it shows that # is significantly smaller than 1, which implies (i) that participants did take into account the anticipated performance of the nonchosen alternative when evaluating the satisfaction associated with a considered alternative, and (ii) that they attached a relatively high weight to small differences in performance between chosen

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Overall regret

2.5 2.0 1.5 1.0 0.5 0.0

00 .15 .30 .45 .60 .75 .90 .05 .20 .35 .50 .65 .80 .95 1 1 1 1 1 1 1 0 0 0 0 0 0 Summed attribute-by-attribure regret

0.

Figure 2. Concave regret (# ˆ 0:654† shown by diamonds versus linear regret (# ˆ 1) shown by squares. # is the concavity ^ convexity parameter.

and nonchosen alternatives while discounting larger differences. Figure 2 shows how the estimated concave regret function transforms summed attribute-by-attribute regret into overall regret. Remember that this summed attribute-by-attribute regret is a measure of the degree to which the nonchosen alternative is better than the chosen one, concerning attributes on which it is better. The linear regret function (implying a reduction to utility maximization) is given as a reference case. We proceed by testing whether the unrestricted regret-based model provides a statistically significant increase in fit compared with the regret-based model where # ˆ 1 (or equivalently, the utility-based model). As is clearly seen, the unrestricted model achieves a substantially higher log-likelihood than the restricted one. However, this increase in model fit should be traded off against the decrease in parsimony (one additional parameter is estimated in the regret-based model). This increase in adjusted r 2 provides a first indication that the increase in model fit more than makes up for the decrease in parsimony. Furthermore, since the utility-based model follows from the regret-based model through parameter restriction, it becomes possible to perform a likelihood-ration test. The w 2 -distributed test-statistic equals 2  [log-likelihood (unrestricted model)ÿlog-likelihood (restricted model)] ˆ 46:4. Given one degree of freedom (the difference between the number of estimated parameters for the two models), the value of the statistic greatly exceeds critical values for any reasonable level of significance (for example, critical value for 1%, significance level ˆ 10:83). As a result, it appears that the unrestricted regret-based model significantly outperforms its utility-based counterpart in terms of goodness of fit. Finally, we provide two additional, information theoretic rather than statistical, measures of model fit that penalize the number of parameters used in the estimation process: Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). Both the AIC-measure and the BIC-measure show that in terms of combined parsimony and model fit, the regret-based model is clearly to be preferred over the utility-based model. In summary, the estimation results suggest that our regret-based model outperforms its utilitarian counterpart and that participants attached relatively high weight to small differences in performance relative to big ones (that is, the regret function is concave). At this point, it should be noted that, in theory, the differences in goodness-of-fit between the regret-based and utility-based models may be the result of nonlinearities in travellers' evaluation of travel times and costs. If such nonlinearities exist, the utilitybased model as we formulated it is not able to capture these. On the other hand, parameter # enables the regret-based model to capture such nonlinearities to some extent. As such, the existence of substantial nonlinearities would limit the fairness of

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the comparison provided above. In order to test whether the superior performance of the regret-based model is not just an artificial result arising from its capability to capture nonlinear attribute valuations, we estimated two additional utility-based models. The first extends the base-case utility model presented in table 4 by estimating three additional parameters: one for (travel time)1=2 , one for (travel costs)1=2 , and one for (travel time  business trip)1=2. None of the estimated parameters appeared significant. The increase in model fit with respect to the base-case utility model is negligible (log-likelihood increased from ÿ1780:0 to ÿ1779:5). The second model extends the base-case utility model by adding one parameter for [(travel time=100)]2, one for (travel costs)2 , and one for [(travel time  business trip)=100]2. The division of the time-related parameters by 100 ensures that the magnitudes of parameter estimates do not differ too much across attributes, since large differences would lead to numerical problems in the estimation procedure. Only the (travel-time)2 parameter turned out to be significant, giving an increase in log-likelihood to ÿ1777:8. Also this second model does not provide a substantial increase in model fit when compared with the base-cap utility model, notwithstanding the use of three additional parameters. As a result, it seems fair to say that the difference in model fit between the regret-based and utility-based model is not simply an artificial result that arises from an ability to capture nonlinearities in travellers' valuation of the attributes of travel alternatives. As a final thought in this empirical analysis, we wish to mention that our findings corroborate one of the pillars of prospect theory (Kahneman and Tversky, 1979), that people tend to display risk-seeking behaviour when faced with losses. In other words, people are inclined to take risks in order to get out of a bad situation. First note that the estimated concave regret function implies that an increase, relative to some starting point, in summed attribute-by-attribute regret implies an increase in overall regret that is smaller than the decrease in overall regret associated with as large a decrease in attribute-by-attribute regret (see figure 2). In other words, the estimated concave regret function suggests that, when faced with uncertainty as to whether a nonchosen alternative performs better than the chosen one in terms of one or more of its attributes, people will tend to display risk-seeking behaviour. Second, where prospect theory uses the status quo in wealth as a reference point, it seems reasonable to conceive the performance of the considered alternative as a reference point in a regret-based context. When faced with losses relative to this reference point (that is, the extent to which a nonchosen alternative performs better than the considered one in terms of one or more of its attributes), people are found to display risk-seeking behaviour. Combining these considerations it appears that, in line with prospect theory, our regret-based estimation results suggest that people display risk-seeking behaviour in the domain of losses. 5 Conclusions and discussion This paper provides an empirical comparison between two perspectives on spatial-choice behavour ö a regret-minimization perspective, which hypothesizes that spatial decision makers anticipate the performance of nonchosen alternatives when evaluating a considered alternative, and the well-known utility-maximization perspective. In order to provide a meaningful statistical comparison between the two approaches, we formulate a regret-minimization model in such a way that it reduces to utility maximization for a given parameter restriction. Estimation results, based on a binary stated mode experiment, show how the regret-based model outperforms its utilitarian counterpart. Furthermore, it is shown that the difference in model fit is not an artificial result that arises from nonlinearities in the valuation of the attributes of choice alternatives. Finally, we show how participants in the experiments have put relatively

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much weight on small differences in performance between the chosen and nonchosen alternative, and tended to discount larger differences. The consistency of our regret-based estimation results (the concave regret function) with the empirically very well-established notion of risk aversion in the domain of losses (see, for example, Fiegenbaum and Thomas, 1988; Hershey and Schoemaker, 1980; Neilson, 1998; Post and Levy, 2005; Weyland, 1996) provides additional credibility to our regret-based model of spatial choice. Some reflections are appropriate at this point. First, we wish to note that this paper does not aim to try to play down the usefulness of utilitarian approaches in a spatialchoice context. However, we do feel that looking at spatial-choice behaviour from different (nonutilitarian) angles and perspectives is likely to deepen our understanding of its underlying choice dynamics and subtleties. Second, it should be noted that the applicability, from the analyst's point of view, of a regret-based approach as formulated here depends partly on the extent to which choice alternatives have attributes in common. This is a direct consequence of the fact that overall regret is conceptualized as a function of the summed attribute-by-attribute regret. The alternatives are evaluated in terms of every common attribute, and regret is added to the alternative that scores worse for a given attribute. However, in a spatialchoice context, alternatives generally have enough attributes in common to enable a regret-based-model application. Furthermore, attributes that the alternatives do not have in common can be treated indirectly, by having them enter the alternative-specific constant. Notwithstanding this, the fact that the utilitarian approach does not rely on the assumption of common attributes can be seen as an advantage in terms of analysis. Third, the application described in this paper concerns a binary-choice context. Please refer to Chorus et al (2008) for a successful application of regret minimization in a multinomial-choice context. The main difference between these two contexts is that in the multinomial context, one needs to specify whether regret is experienced with respect to all available alternatives that perform better than the one considered, or only the best performing one. It is discussed, using a result obtained by Quiggin (1994), that only the latter specification is consistent with the requirement of irrelevance of statewise dominated alternatives. This requirement states that a choice from a given choice set should not be affected by adding to, or removing from, this set an alternative that is dominated by the other alternatives. Apart from this, there is no difference between the binary-model and multinomial-model specification. Finally, we again wish to stress that our sample does not constitute a random sample, and is not representative of the general population of travellers; also, our analysis involves choices in a hypothetical context. Empirical work based on random samples and actual spatial-choice situations (allowing for the collection of revealed choice data) is needed to provide further evidence of the usefulness of regret-minimization models in a spatial-choice context. In summary, we consider that our findings suggest that our model of random regret minimization is a promising candidate for spatial-choice analysis, and a useful alternative to random utility-maximization approaches: firstly, it provided a natural formalization of the intuition that the anticipated performance of nonchosen alternatives relative to the considered alternative influences choice behaviour. Secondly, the model's concavity ^ convexity parameter provides an elegant means to estimate how decision makers evaluate small versus large differences in performance between alternatives. Thirdly, in the context of the dataset considered in this paper, the regret-based model performs better than the best possible random utility specification in terms of goodness of fit.

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