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If you choose not to decide, you still have made a choice
Journal of Choice Modelling 22 (2017) 13–23
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Journal of Choice Modelling journal homepage: www.elsevier.com/locate/jocm
If you choose not to decide, you still have made a choice Francisco J. Bahamonde-Birke
a,b,c,⁎
d
, Isidora Navarro , Juan de Dios Ortúzar
MARK e
a
Institut für Verkehrsforschung, Deutsches Zentrum für Luft- und Raumfahrt (DLR), Germany b Energy, Transportation and Environment Department, Deutsches Institut für Wirtschaftsforschung, Berlin, Germany c Technische Universität Berlin, Germany d Department of Transport Engineering and Logistics, Pontificia Universidad Católica de Chile, Chile e Department of Transport Engineering and Logistics, Centre for Sustainable Urban Development (CEDEUS), Pontificia Universidad Católica de Chile, Chile
A R T I C L E I N F O
ABSTRACT
Keywords: Discrete choice models Indifference Stated-choice experiments
When designing stated-choice experiments modellers may consider offering respondents an “indifference” alternative to avoid stochastic choices when utility differences between alternatives are perceived as too small. By doing this, the modeller avoids adding white noise to the data and may gain additional information. This paper proposes a framework to model discrete choices in the presence of indifference alternatives. The approach allows depicting the likelihood function, independent of the number of alternatives in the choice-set and in the subset of indifference alternatives, offering a new approach to existing methods that are only defined for binary choice situations. The method is tested with the help of simulated and real data observing that the proposed framework allows recovering the parameters used in the generation of the synthetic datasets without major difficulties in most cases. Alternative approaches, such as considering the indifference option as an opt-out alternative or ignoring the indifference choices are clearly outperformed by the proposed framework and appear not capable of recovering parameters in the simulated set.
1. Introduction Discrete choice models rely on the assumption that individuals are rational decision makers that maximize their utility when facing a given choice situation. This way, individuals will opt for a given alternative if and only if it promises them the maximum expected utility among all alternatives in their choice-sets (Thurstone, 1927; McFadden, 1974). Nevertheless, establishing which alternatives should be considered into the individuals’ choice-set is not an easy task. In real situations the modeller will just observe the chosen alternatives and needs to construct the choice-sets of the individuals on the basis of their characteristic and their choices. This involves major difficulties, as it is well established that people tend to narrow their decisions to only a subset of the potentially available options (Roberts and Lattin, 1991; Swait and Erdem, 2007). By contrast, when dealing with stated preference (SP) data the choice-set must be established a priori. In this case, it is important that it be carefully defined preserving the realism of the choice situations. Thus, in many cases it might be necessary to consider an opt-out (non-purchase) alternative (Carson et al., 1994; Olsen and Swait, 1998); whether this alternative should be included directly into the choice-set (Louviere et al., 2000) or indirectly via dual response procedures (Dhar and Simonson, 2003) remains a debatable point (see Schlereth and Skiera (2016) for a good discussion), but the necessity of alternatives accounting for a reservation utility level
⁎
Corresponding author at: Institut für Verkehrsforschung, Deutsches Zentrum für Luft- und Raumfahrt (DLR), Germany. E-mail addresses: [email protected], [email protected] (F.J. Bahamonde-Birke), [email protected] (I. Navarro), [email protected] (J.d.D. Ortúzar). http://dx.doi.org/10.1016/j.jocm.2016.11.002 Received 23 February 2016; Received in revised form 23 November 2016; Accepted 26 November 2016 1755-5345/ © 2016 Published by Elsevier Ltd.
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higher than the expected utility of all alternatives in the choice-set is well-established (Kontoleon and Yabe, 2003). A similar but far less analysed problem is the inclusion of indifference alternatives in the choice-set. If the modeller does not allow for respondents to state their indifference among two or more alternatives, they will be forced to opt for one of them in a rather stochastic manner, adding white noise to the experiment. Additionally, doing so would provide less information about the individuals’ preferences leading to loss of efficiency. Indeed, Cantillo et al. (2010) used a synthetic dataset to show that assigning the preferences associated with an indifference alternative randomly, diminished significantly the model's capability to recover the input parameters. Furthermore, Cantillo et al. (2010) also considered real databanks, observing than offering the possibility of stating indifference may indeed affect the outcome of the experiment (the estimated parameter values). Along these lines, empirical evidence (Dhar, 1997; Fenichel et al., 2009) shows that in experiments including non-purchase options, indifference situations may artificially increase the probability of selecting the opt-out1 alternative, as a kind of cognitive bias. Therefore, it appears advisable to include indifference alternatives when also offering non-purchase options as a way to reduce cognitive biases. Nevertheless, including indifference alternatives should be carefully considered, as it might generate other kind of complications, especially if individuals are overwhelmed by the complexity of the choice experiment. Both situations (opt-out and indifference alternatives) exhibit, however, substantial differences; while the former suggest the existence of a reservation utility that is higher than the utility provided by the alternatives in the choice set, the latter indicates that individuals ascribe the same utility to two or more alternatives in it (this utility being higher than the reservation utility). Therefore, in the first case an extra alternative accounting for this reservation prize should be considered. Nevertheless, considering an extra alternative to reflect indifference choices does not seem to be appropriate, as it does not reflect the causes leading to the statement of indifference; in fact, by treating indifference as a new (opt-out) alternative, the modeller implicitly assumes that the utility ascribed to this new option would be greater than that of the competing alternatives, which is clearly not the case. Despite the fact, that according to classical theory indifference situations will only arise if the expected utility of two or more alternatives is the same (curves of indifference), the underlying behavioural theory behind the indifference phenomena suggests the existence of perception thresholds, below which the individuals are not able to perceive differences between two stimuli (Quandt, 1956; Coombs et al., 1970; Cantillo and Ortúzar, 2006). Krishnan (1977) developed an operational discrete choice model accounting for the existence of indifference thresholds. This approach (Minimum Perceivable Differences model, MPD) allows taking into account the fact that observations falling into the indifference interval would be assigned stochastically to one alternative, in the context of a binary choice situation. Cantillo et al. (2010) expanded the MPD-approach to allow for individuals stating their indifference in stated-choice (SC) experiments. This way, the indifference alternative would be selected if the difference between the utility of both alternatives was smaller than a threshold, to be estimated. The main limitation of the MPD-approach is that it only allows considering binary choice situations. Thus, the method can neither consider situations where two alternatives exhibit a similar utility (which is superior to all other alternatives in the choiceset), nor cases when three or more choices report an apparently identical utility (which may be of particular interest when considering alternatives to first-choice SP experiments, such as rankings). The same limitation arises, when considering approaches such as an ordered logit framework (with indifference being an intermediate choice between two binary alternatives). A method that allows accommodating more than two alternatives, consists in assuming that instead of behaving as utility maximizers, the individuals minimize their regret (RRM framework; Chorus, 2012a). Under this assumption, the regret associated with a certain alternative is given by the direct comparison of its attributes with those of all remaining alternatives in the choice set (whereby only a negative performance would generate regret). Thus, including an extra null-alternative (without attributes) into the model cannot longer be associated with a higher reservation utility, but would rather stand for a level of regret, above which none of the alternatives in the choice-set is favoured (i.e. none of the alternatives significantly minimizes the regret in comparison with other options in the choice-set, Chorus, 2012b). Hence, under this assumption an extra alternative would allow to capture indifference (Hess et al., 2014). Nevertheless, this approach does not appear appropriate to deal with non-binary choice situations, as the extra alternative would be indicative of indifference among the whole choice-set, not allowing to consider indifference among a sub-set of alternatives (e.g. the respondent is indifferent among two alternatives, but both alternatives are preferred over the remaining alternatives in the choice-set). Furthermore, it does not seem appropriate to consider experiments allowing for both the option of stating indifference as well as opting out, because: (i) it would be necessarily a non-binary situation, and (ii) it would imply considering two different nullalternatives, which should account for two completely different phenomena. Finally, the approach would necessarily require the analyst to assume a regret minimization strategy (ideally, the modeller should aim at an approach that allows considering indifference under different assumptions, e.g. regret minimization or utility maximization, and discern between them on an empirical basis). This paper discusses the implications of indifference choices concerning the utility ascribed to the different alternatives in the choice-set. Along these lines, the paper presents a new approach that allows dealing with indifference choices in multinomial choice situations. This framework allows not only addressing first-choice SP experiments but also rankings, where indifference choices may be expected to appear more often. The approach is tested with the help of simulated and real datasets, observing that it clearly
1 For the purposes of this paper, it is assumed that opting-out implies that none of the alternatives satisfies the individual's requirements (i.e. a non-purchase option). This is highly recommended in non-pivoted SP experiments, as with totally new options it could well be that none is acceptable and not presenting an opt-out may bias results (Olsen and Swait, 1998).
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outperforms other methods allowing for a proper estimation of parameters. 2. Modelling with indifference alternatives According to random utility theory (Thurstone, 1927; McFadden, 1974) in a given choice situation an individual q would opt for alternative i belonging to choice-set A if and only if Uiq > Ujq ∀ j ≠ i ∈ A. Then, the probability with which the individual would select this alternative is given by:
Piq = P(Uiq > Ujq )
∀j≠i∈A
(2.1)
The expected utility U can be expressed in terms of a representative component V, characterized through concrete and measurable properties of the alternatives and the individuals, and an error term ε representing all unknown (for the analyst) elements of the decision. Assuming an additive specification (other structures are also possible, but for didactic purposes we assume the more common specification), (2.1) can be rewritten as:
Piq = P(Viq + εiq > Vjq + εjq ) = P(Viq − Vjq > εjq − εiq )
∀j≠i∈A
(2.2)
Finally, the likelihood of observing a set of given choices is given by:
L=
∏ ∏ Pjq yjq, q
(2.3)
j∈A
where yjq takes the value of one if alternative j is selected by individual q and zero otherwise. 2.1. Stating indifference If the modeller allows for individual q to state his indifference between n alternatives belonging to an indifference-set B, which is a subset of the complete choice-set A, (consisting of m alternatives, with m≥n), the choice probability of the alternatives belonging to B, may be written as follows:
Pi = Pj = Pk = …
∀ i, j, k , … ∈ B
(2.4)
As under random utility theory all alternatives belonging to B must maximize the individual's expected utility U, Eq. (2.4) can be rewritten in these terms:
Ui ≈ Uj ≈ Uk ≈ …
∀ i, j, k , … ∈ B,
(2.5)
where the inequality accounts for the existence of utility differences (smaller than a given threshold) that are not being perceived by the respondents. If we introduce error terms ϕ, then Eq. (2.5) can be expressed in terms of equalities (where ϕ is an error term merely guaranteeing that the utility differences within a given indifference-set be equal to zero):
Ui + ϕi = Uj + ϕj = Uk + ϕk = …
∀ i, j, k , … ∈ B
(2.6)
Then, expressing the expected utility in terms of a representative utility V and the aforementioned error terms ε (accounting for all unknown elements of the decision but the indifference thresholds leading to the statement of indifference) leads to the following expression:
Vi + εi + ϕi = Vj + εj + ϕj = Vk + εk + ϕk = …
∀ i, j, k , … ∈ B
(2.7)
Finally, the probability associated with an alternative in the indifference set would be given by:
Pri = P(Vi − Vj > εj + ϕj − εi − ϕi )
∀ j ≠ i ∈ A ∧ ϕj = 0, ∀ j ∉ B
(2.8)
where Pri≠Pi, as it is associated with a larger error component (due to ϕ). When modelling with indifference alternatives, it cannot be assumed that all alternatives in the indifference-set are selected at the same time, but rather that all are selected with a frequency 1/n. Hence, when considering indifference options, the likelihood of observing a given choice can be expressed as:
L=
∏ Prj yj, (2.9)
j∈A
where yj takes the value of 1/n if alternative j∈B and zero otherwise. If we consider a group of individuals M selecting indifferencesets and another group of individuals L selecting unique preferred alternatives (where M and L are mutually exclusive), the general likelihood function for the population will take the following form:
L=
∏ ∏ Pjq yj⋅ ∏ ∏ Prjq yj, q∈M j∈A
(2.10)
q∈L j∈A
It is important to notice that using a value of yj equal to one for all alternatives in an indifference-set would result in overweighting all observations by respondents selecting indifference options. Also, note that the likelihood function would be maximized when all alternatives in an indifference-set are equally likely (AM-GM inequality). Even though, the above framework was derived for utility maximization, it is straightforward to extend it to consider regret 15
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minimization strategies (Chorus, 2012a). Furthermore, as the meaning of null-alternatives is different when pursuing regret minimization and utility maximization strategies, this approach would allow for a fair comparison between both sets of assumptions in the presence of indifference. This would allow for the modeller to discern which set of assumptions offers a better representation of the underlying decision-making process on the basis of the empirical evidence. Similarly, it is easy to extend it for ranking preference elicitation procedures when two or more alternatives at different depths of the ranking order are considered equivalent (i.e. the respondent cannot choose among them). In this case, the modeller would rely on the choices arising by exploiting the ranking2 (Chapman and Staelin, 1982; Bradley and Daly, 1994) and the framework would apply to those cases where there is indifference between two or more alternatives (i.e. allowing to deal with ties at any depth of the ranking order). 2.2. Identification of the error term ϕ Identification and the distribution of the error terms ϕ are complicated topics. First, the errors only appear in the utility function if the alternatives are selected as part of an indifference-set (as part or Pri), which may significantly decrease the number of observations available for estimation. Furthermore, this additional error term will only be associated with the alternatives that are indeed part of a given indifference set; thus, the utility functions of the remaining alternatives (not selected as a part of the indifference set) will not be affected by this term. For this reason, non-symmetrical assumptions regarding its distribution are unsuitable (the estimator would diverge, as it would only be associated with selected alternatives). Moreover, establishing an adequate functional form for the error term ϕ is not an easy task, as it does not comply with the usual assumptions regarding error terms. In fact, ϕ is an error component that merely guarantees that the difference among two or more different utilities is equal to zero; that is, the error is being added to the alternatives associated with the lesser utilities or subtracted from the alternatives associated with higher utilities, so that the utility of all alternatives in the indifference-set add up to the same amount. Hence, these errors would not be properly represented by the usual assumptions concerning error terms. That, in conjunction with the aforementioned unsuitability of non-symmetrical distributions, creates major difficulties. If the modeller assumes that the error term ϕ follows a distribution equal to the difference between two Logistic distributions with different scale parameters,3 the sum εi − εj + ϕ would also be represented by a Logistic distribution, with a smaller scale parameter (i.e. with a larger standard deviation). As a consequence, the utility functions of alternatives being selected as a part of an indifference-set would be equivalent to the utility functions of the single-choice framework (Pi), but would be associated with a smaller scale parameter, reflecting the increased uncertainty.4 A limitation of this approach is that it does not allow quantifying the thresholds leading to the statement of indifference; notwithstanding, the model stills offers an adequate depiction of the underlying utility functions associated with the different alternatives as well as a functional form for forecasting. Additionally, the approach allows considering different scale parameters depending on the number of alternatives in the indifference-sets, or on which alternatives are being considered as part of the indifference-sets, as well as on the socio-economic characteristics on the individuals. Because of identification issues, the modeller is forced to normalize one of the aforementioned scale parameters (either that related to the equations for single choices – when the respondent selects a unique alternative - or that associated with alternatives selected as part of an indifference-set). Both normalizations would lead, however, to different estimates; if the modeller normalizes the error associated with the single choices, s/he would be scaling the estimates upwards (when comparing them with the outcomes of an alternative model, where the scale parameter was fixed at one for the whole sample), as single choices are related to a lesser degree of uncertainty (i.e. the differences of representative utilities are larger, in average, when indifference situations are left out or alternatively, indifference situations would arise more often when utility differences are smaller). Thus, when fixing the scale parameter for single choices the estimates are no longer comparable with alternative specifications.5 In this case, the analyst does not attempt to estimate separate models for individuals choosing single alternatives or indifference sets, but rather to estimate a parsimonious model that is consistent with both kind of choices (and the estimates of which are comparable with models estimated without introducing the error term ϕ). A possible way to deal with this problem is to fix the scale parameter associated with the weighted average of the sample (normally at one, but other values may be chosen). In this form, the analyst would estimate k-1 parameters (with k representing the number of different scale parameters), and express the last one in terms of the others to be estimated as in:
λ⋅N = λ1⋅n1 + λ 2⋅n 2 + λ 3⋅n3 + …,
(2.11)
where N represents the total number of observations in the sample, λ the weighted average of the scale parameters, λj scale parameters to be estimated, and nj the number of observations associated with a given scale parameter. A final characteristic of the approach is that when considering only two alternatives, different scale parameters cannot be 2 By exploiting the ranking the analyst assumes that the first alternative (in the ranking) would be selected in a choice-set considering all alternatives. The second ranked alternative would be the choice in a set including all alternatives but the first ranked, and so on. 3 While this assumption does not appear to be wholly appropriate to describe the aforementioned response, it evidences itself as fairly convenient for modelling issues. 4 This framework resembles the structure used to deal with RP and SP data simultaneously. 5 Naturally, as the scale parameters are fixed without loss of generality, the model would still be appropriate, but a correction would be required in order to make different specifications comparable.
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estimated. This relates to the fact that when all alternatives are equally probable, all estimates must necessarily be equal to zero and therefore any scale parameter multiplying a group of indifference sets containing all alternatives would be unidentified (and therefore can be excluded without loss of generality). As in binary choice situations every indifference-set must necessarily include both alternatives, considering different scale parameters in this case would be redundant. As a corollary, considering different scale parameters would only impact the model when those can be associated with at least one indifference-set (selected by at least a part of the population) not containing all alternatives in the choice-set. 3. Study cases To test our framework as well as the impact of introducing indifference alternatives into the model, we conducted four different empirical tests. First, we considered a simulation exercise consisting of only two alternatives (binary choice situation). Then, we analysed a real case considering a similar binary structure. Then, a second simulation exercise introduced an opt-out alternative (additional to the possibility of stating indifference), and finally, we tested the framework using real data consisting of a binary choice set allowing for indifference as well as for opting out. 3.1. Simulated binary choices Our first simulated scenario considered binary choice situations where the utility functions of the alternatives were given by:
U1 = β1⋅X1 + ε1 U2 = ASC2 + β2⋅X2 + ε2
(3.1)
Here, individuals are supposed to choose the alternative with the highest expected utility or to state their indifference if the utility difference between both alternatives is smaller than a given threshold. We took X1 and X2 as random draws from Normal distributions with mean 0 and −1, and standard deviations 1 and 1.5, respectively. The error terms were assumed to be independently EV1 distributed with mean 0 and scale parameter 1; both β parameters as well as the ASC were fixed to 1, while three different values were tested for the threshold: 0.15, 0.3 and 0.45 representing small, medium and large indifference intervals, respectively. In all three cases 5000 pseudo-individuals were generated observing that 254, 495 and 739 stated their indifference between both alternatives, respectively. With this dataset we estimated four models, considering different approaches to address the indifference alternatives. Model 1 considered all indifference options as an extra alternative; thus, in our utility maximization framework this extra alternative represents opting-out rather than indifference.6 Model 2 presents the results for a case where the individuals stating their indifference are ignored. Model 3 does not account for the fact that all alternatives in a selected indifference-set are not observed with the same frequency (if a given alternative is selected as a part of a binary indifference set, in reality, it would only be selected with a frequency of 0.5) as alternatives being selected directly; hence, Model 3 assigns a weight (yj) of 1 to all observations. Finally, Model 4 considered the proposed framework (Eq. (2.10)). All models were estimated using PythonBiogeme (Bierlaire, 2003), and the results are presented in Table 3.1 (the standard deviation of the estimated parameters is shown in parenthesis). As can be observed, considering the indifference options as opt-out alternatives yields a significantly worse goodness-of-fit than the other approaches. Nevertheless, in this particular case the log-likelihood does not offer meaningful insights, as Model 1 considers one alternative more (affecting the choice probabilities), while Models 2 and 3 consider less/more observations (ignoring/ overweighting the observations associated with smaller utility differences would artificially diminish/increase the error and goodness-of-fit). An analysis of the parameters provides more useful information. Models 1–3 are not capable of recovering the values used in the generation of the dataset (the t-tests of equality are rejected at a significance level of 5%, for at least part of the estimators in every case). This may be related to the fact that the artificial increase/descent of the error affects the scale parameter, biasing the results. Model 4 (the proposed framework) recovers the parameters used in the generation of the dataset without major difficulties. Regarding the indifference intervals, it can be observed that the differences between the approaches increase (as well as the difficulties associated with the parameter recovery for Models 1–3) as the indifference threshold gets greater leading to more individuals stating their indifference. Model 4 performs adequately in every case. 3.2. Real dataset: binary choices The second study case consists of three different datasets that were collected as a part of the same survey on passenger rail transportation in Southern Chile (Appendix A presents an example of the way alternatives were presented to the individuals). The different datasets are associated with three different kinds of trips: short and long interurban trips and urban trips. In this regard, the individuals were asked to choose between two unlabelled hypothetical alternatives, having also the choice of stating their indifference among both alternatives. All experiments considered the same attributes: fare (P), travel time (TT), access time (AT) and 6 As we are considering a binary choice situation in this specific case, it would have been possible to test the approach put forward by Hess et al. (2014) by assuming regret minimization. Nevertheless, such comparison would be spurious as we assumed utility maximization when constructing the simulated dataset; the same applies (though to a lesser extent) to alternatives methods, such as the MPD-approach (Cantillo et al., 2010).
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Table 3.1 Estimated models. Case study 1. Threshold
Variable
Model 1
Model 2
Model 3
Model 4
0.15
X1 X2 ASC2 ASCopt-out
0.970 (0.0392) 0.964 (0.0307) 0.905 (0.0454) −2.30 (0.0679)
1.10 (0.0445) 1.07 (0.0345) 1.04 (0.0503) –
0.973 (0.0397) 0.950 (0.0304) 0.909 (0.045) –
1.03 (0.0419) 1.01 (0.0323) 0.968 (0.0474) –
Final log-likelihood No. of observations
−3248.472 5000
−2190.648 4746
−2626.479 5254
- 2410.844 5000
X1 X2 ASC2 ASCopt-out
0.897 (0.0379) 0.969 (0.0304) 0.933 (0.0451) −1.55 (0.0516)
1.14 (0.0430) 1.16 (0.0344) 1.19 (0.0489) –
0.904 (0.0381) 0.933 (0.0294) 0.934 (0.0438) –
1.00 (0.0401) 1.03 (0.0314) 1.04 (0.0459) –
Final log-likelihood No. of observations
−3693.051 5000
−1978.882 4505
−2812.637 5495
- 2403.098 5000
X1 X2 ASC2 ASCopt-out
0.846 (0.0366) 0.906 (0.0294) 0.872 (0.0441) −1.05 (0.0445)
1.15 (0.0484) 1.15 (0.0383) 1.18 (0.0561) –
0.819 (0.0353) 0.811 (0.0267) 0.834 (0.0413) –
0.945 (0.0402) 0.941 (0.0309) 0.965 (0.0468) –
Final log-likelihood No. of observations
−4082.823 5000
−1868.977 4261
−3109.345 5739
- 2505.769 5000
0.3
0.45
transport mode (TM – indicating if the trip was made by train or by bus). We considered the following utility functions:
U1 = ASC1 + βP ⋅P1 + βTT ⋅TT1 + βAT ⋅AT1 + βTM ⋅TM1 + ε1 U2 = ASC2 + βP ⋅P2 + βTT ⋅TT2 + βAT ⋅AT2 + βTM ⋅TM2 + ε2
(3.2)
The first dataset (short interurban trips) consists of 2781 observations, of which 289 (10.4%) stated indifference among both options. The second dataset (long interurban trips) includes 2745 observations, with 202 (7.4%) indicating indifference. Finally, the data on urban trips consists of 5841 observations, 553 (9.5%) of which were associated with indifference. For consistency purposes we considered basically the same models as in the previous study case. This way, Model 1 considers indifference as an additional opt-out alternative, Model 2 ignores all indifference observations, Model 3 relays on the proposed approach without accounting for frequency considerations (in fact, duplicating the weight of the indifference observations) and Model 4 follows the proposed approach, as depicted in Eq. (2.10). For illustrative purposes, this time we also included a model considering a null-alternative in the context of a regret minimization framework (Model 5). In this case, the additional alternative, can be framed as indifference (Hess et al., 2014).7 The results are presented in Table 3.2. In line with the previous study case, the results confirm that treating indifference as an opt-out alternative (in the utility maximization framework) lead to results that widely diverge from alternative approaches (however, it is not possible to establish positively that the results are definitely biased, as it is not possible to observe the underlying model). Along these lines, as in the previous case, the estimates of Model 2 and Model 3 are consistently deviated up and down, respectively, from the results associated with Model 4. All three considered cases (interurban short and long trips, as well as urban trips) exhibit exactly the same patterns. Finally, and according with the expectations, the estimates associated with Model 5 are similar, but slightly larger, than those obtained for Model 4 (with the exception of the ASCs, as they are indicative for utility in RUM models and regret in RRM models). However, a direct comparison between both models cannot be performed, as they consider a different number of alternatives (although, surprisingly, Model 5 exhibits a worse goodness-of-fit than Model 1 for urban trips). These results confirm our previous findings, implying that treating indifference as an opt-out alternative, as well as ignoring indifference observations or applying the proposed approach without considering frequency issues may lead to biased results. This evidence supports the necessity of an adequate modelling approach when dealing with indifference observations and real data. 3.3. Simulated binary choices including an opt-out alternative A second simulated scenario was considered. The dataset was generated similarly to the previous one, but in this case we introduced an opt-out alternative,8 as shown in the following equations set. 7 It could have also been possible to consider Models 2–4 on the basis of regret minimization, but given that they are binary cases, regret minimization and utility maximization would lead to the same outcome. 8 Even though we have framed the third alternative as opting-out, it is easy to see that, for the purposes of the modelling, it would be equivalent to considering a choice-set with three alternatives.
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Table 3.2 Estimated models. Case study 2. Survey
Variable
Model 1
Model 2
Model 3
Model 4
Model 5
Short Trips
P TT AT TM ASC2 ASCopt-out
−0.83 (0.204) −0.172 (0.0371) −0.067 (0.0546) 0.148 (0.0788) −0.493 (0.0779) −4.06 (0.45)
−1.99 (0.343) −0.375 (0.0662) −0.0778 (0.0673) 0.272 (0.0878) −0.507 (0.128) –
−1.63 (0.301) −0.299 (0.0596) −0.0429 (0.0592) 0.202 (0.0781) −0.402 (0.114) –
−1.79 (0.32) −0.332 (0.0626) −0.0571 (0.0628) 0.232 (0.0825) −0.447 (0.12) –
−1.97 (0.337) −0.351 (0.0635) −0.0667 (0.0684) 0.258 (0.0869) 0.482 (0.125) 4.5 (0.0879)
log-likelihood No. of obs.
2781.00 2781
−1620.24 2492
−2041.62 3070
−1832.10 2781
− 2549.21 2781
P TT AT TM ASC2 ASCopt-out
−0.855 (0.161) −0.112 (0.0342) −0.112 (0.0534) 0.232 (0.0792) 0.475 (0.0964) −6.17 (0.81)
−1.06 (0.202) −0.344 (0.0508) −0.123 (0.066) 0.268 (0.0876) 0.0866 (0.137) –
−0.876 (0.185) −0.278 (0.0465) −0.102 (0.0601) 0.264 (0.0812) 0.0769 (0.126) –
−0.957 (0.192) −0.308 (0.0485) −0.111 (0.0628) 0.267 (0.0841) 0.0807 (0.131) –
−0.988 (0.194) −0.352 (0.0513) −0.126 (0.0668) 0.28 (0.0866) −0.0428 (0.137) 4.64 (0.108)
log-likelihood No. of obs.
−2403.90 2745
−1666.02 2543
−1960.91 2947
−1814.13 2745
−2387.72 2745
P TT AT TM ASC2 ASCopt-out
−1.1 (0.263) −0.452 (0.0432) −0.169 (0.0659) 0.3 (0.052) −0.318 (0.0733) −3.38 (0.187)
−1.25 (0.273) −0.427 (0.0455) −0.178 (0.0689) 0.277 (0.059) −0.257 (0.0763) –
−1.01 (0.246) −0.353 (0.0406) −0.146 (0.0616) 0.236 (0.0533) −0.217 (0.068) –
−1.12 (0.258) −0.386 (0.0428) −0.16 (0.0649) 0.255 (0.0559) −0.235 (0.0717) –
−1.2 (0.268) −0.417 (0.0443) −0.172 (0.0673) 0.284 (0.059) 0.261 (0.0745) 4.49 (0.0582)
log-likelihood No. of obs.
−5408.35 5841
−3589.52 5288
−4368.88 6394
−3979.80 5841
−5419.95 5841
Long Trips
Urban Trips
U1 = β1⋅X1 + ε1 U2 = ASC2 + β2⋅X2 + ε2 Uopt−out = ASCopt − out + ε3
(3.3)
The data generation process was analogous to the previous section, but this time the pseudo-individuals were allowed to state their indifference among: (i) both alternatives, (ii) one existing alternative and opting-out, and (iii) both existing alternatives and opting-out. The ASC associated with the opt-out alternative (accounting for the reservation utility) was fixed at zero. Again we considered the same different indifference thresholds. In the first case (low threshold), we observed 400 observations indicating indifference (124 between both existing alternatives, 131 between alternative 1 and opting out, 135 between alternative 2 and opting out, and 10 between both alternatives and opting out). In the second case (intermediate indifference threshold) 745 pseudo-individuals stated their indifference (200 between both existing alternatives, 260 between alternative 1 and opting out, 230 between alternative 2 and opting out, and 55 between both alternatives and opting out), while in the third case (large indifference threshold) we observed 1093 indifference observations (297 between both existing alternatives, 345 between alternative 1 and opting out, 319 between alternative 2 and opting out, and 132 between both alternatives and opting out). In this case, considering extra alternatives does not make sense, as option 3 already accounts for the opt-out alternative. This way, we estimated four different models. Model 1 presents the results of ignoring the indifference choices. Model 2 considers the proposed framework but ignores the error term ϕ and does not consider a different scale parameter for the observations associated with indifference-sets. Model 3 considers different scale parameters but assigns a weight (yj) of 1 to all observations. Finally, Model 4 represents the proposed framework. The scale parameters were fixed in accordance with (2.10) and (2.11). The results are presented in Table 3.3. This time, all specifications but Model 1 (which ignores the indifference observations) allow for an adequate estimation of the parameters used in the generation of the dataset. Model 2 performs surprisingly well, but still offers a significantly worse goodnessof-fit than Model 4. Nevertheless, the fact that Model 2 performs acceptably while ignoring the different variability, suggest that considering different scale parameters may be omitted without major complications if the modeller does not count with enough indifference-set observations (to allow for a correct estimation of the λ parameter). Model 3 performs significantly better than its un-weighted counterpart in the previous simulation exercise. The reason is that when considering different scale parameters, the weighting only affects the observations multiplied by the same scale parameter. Then, in this case, the weights associated with the single choices are not relevant, while we observed a small amount of observations with a weight of 1/3 (individuals stating their indifference among all choices) and a large majority associated with a weight of 1/2, which clearly diminishes the bias associated with ignoring the weighting. In fact, if we ignore the scale parameter (analogously to 19
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Table 3.3 Estimated models. Case study 3. Threshold
Variable
Model 1
Model 2
Model 3
Model 4
0.15
X1 X2 ASC2 ASCopt-out λIndifference-set
1.09 (0.0430) 1.05 (0.0334) 1.11 (0.0493) 0.0242 (0.0419) –
1.03 (0.0404) 0.991 (0.0310) 1.05 (0.0464) 0.0272 (0.0396) –
1.04 (0.0401) 0.999 (0.0310) 1.05 (0.0461) 0.0283 (0.0391) 0.436 (0.0511)
1.04 (0.0406) 0.999 (0.0314) 1.05 (0.0466) 0.0257 (0.0395) 0.439 (0.0717)
Final log-likelihood No. of observations
3848.233 4600
−4294.822 5000
−4699.123 5410
4268.096 5000
X1 X2 ASC2 ASCopt-out λIndifference-set
1.08 (0.0441) 1.08 (0.036) 1.08 (0.0514) 0.0394 (0.0431) –
0.976 (0.0391) 0.963 (0.0311) 0.996 (0.0461) 0.0304 (0.0388) –
0.979 (0.0383) 0.968 (0.0309) 0.989 (0.0449) 0.0302 (0.0377) 0.438 (0.0385)
0.98 (0.0393) 0.974 (0.0318) 0.981 (0.046) 0.0355 (0.0383) 0.447 (0.0535)
Final log-likelihood No. of observations
3572.903 4255
−4404.245 5000
−5202.828 5800
−4356.509 5000
X1 X2 ASC2 ASCopt-out λIndifference-set
1.17 (0.0468) 1.13 (0.0383) 1.19 (0.0559) 0.0242 (0.0419) –
0.979 (0.0386) 0.776 (0.0272) 1.00 (0.0452) −0.0785 (0.0398) –
0.937 (0.0383) 0.858 (0.0291) 0.955 (0.0448) 0.0291 (0.0366) 0.153 (0.0355)
0.933 (0.0390) 0.872 (0.0298) 0.941 (0.0456) 0.0267 (0.0362) 0.142 (0.0479)
Final log-likelihood No. of observations
−3163.748 3907
−4515.938 5000
−5701.262 6225
−4360.298 5000
0.3
0.45
Model 2 but ignoring the weighting), the model is no longer capable of recovering the parameters. Regarding the magnitude of the threshold we observe that all specifications (besides Model 1) perform adequately as long as the indifference threshold is small or intermediate. When the indifference threshold is large (causing that an important part of our pseudo-population is associated with indifference alternatives), all models exhibit some difficulties recovering the parameters. This may be explained by the fact that a larger indifference threshold may be indeed associated with a larger uncertainty and that our assumptions regarding the distribution of the error term ϕ are not exactly accurate. Even in this case, Model 4 exhibits the best performance, and is capable of recovering three out of four parameters and clearly outperforms Model 1 and Model 2. 3.4. Real dataset including opt-out and indifference options For the fourth case study we used another real dataset. Information was collected as part of a project aiming to identify and assess the value of urban attributes (Appendix B presents an example of the way alternatives were presented to the individuals). For this purpose, a SP experiment in the context of residential location choice was designed and conducted in one neighbourhood of Santiago de Chile. The area has some special characteristics, such as high income, urban amenities and enhanced possibilities (in terms of transport facilities). The attributes considered in the experiment were the existence of Green Areas on the sidewalk (GA), Bus Corridors (BC), Green Verges segregating the bus corridors (GV), Bike Lanes (BL), and changes in the housing rent (ΔP). The experiment offered respondents four different options; two alternatives depicted through visual representations (with utilities U1 and U2), the possibility of stating indifference between both of them, as well as an opt-out alternative (with utility Uopt-out), as in the following equations:
U1 = ASC1 + βga⋅GA1 + βgv⋅GV1 + βbike⋅BL1 + βbus⋅BC1 + βprice⋅ΔP1 + ε1 U2 = ASC2 + βga⋅GA2 + βgv⋅GV2 + βbike⋅BL 2 + βbus⋅BC2 + βprice⋅ΔP2 + ε2 Uopt−out = ASCopt − out + ε3
(3.4)
The dataset has 588 observations, twelve of which (2.04%) are associated with indifference statements, and 26 with the opt-out alternative. We estimated four different models (Table 3.4) similar to those described in the third case study. Model 1 ignores the individuals stating their indifference. Model 2 does not consider different scale parameters for the observations associated with the indifference sets, while Model 3 adds different scale parameters but considers the same weight for all observations; finally, Model 4 uses the proposed framework (different scale parameters and different weights). As can be observed, in this case all estimated models perform similarly; in fact, it is not possible to identify statistically significant differences between the parameters estimated by each model. This may be related to the fact that only twelve individuals (2.04%) stated their indifference among alternatives. It implies that considering involved specifications is not crucial when the number of 20
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Table 3.4 Estimated models. Case study 4. Variable
Model 1
Model 2
Model 3
Model 4
GA GV BL BC ΔP ASC1 ASC2 λindifference-set
0.204 (0,112) 0,392 (0.138) 0.977 (0.120) −0.133 (0.130) −1.97 (0.407) 1.60 (0.235) 1.54 (0.236) –
0.196 (0.110) 0.392 (0.136) 0.956 (0.118) −0.129 (0.128) −1.93 (0.400) 1.64 (0.233) 1.58 (0.235) –
0.189 (0.109) 0.391 (0.134) 0.937 (0.118) −0.126 (0.127) −1.89 (0.394) 1.68 (0.232) 1.62 (0.234) 0.977 (0.344)
0.196 (0,110) 0,391 (0.136) 0.957 (0.119) −0.129 (0.128) −1.93 (0.400) 1.64 (0.234) 1.58 (0.235) 0.967 (0.474)
Final log-likelihood No. of observations
−438.742 576
−448.481 588
−458.165 600
−448.478 588
indifference observations is relatively low. In line with the previous case study it is not possible to identify differences associated with the inclusion of different scale parameters for the indifferent sets, which supports the hypothesis that the more involved approach does not perform substantially better than assuming equal utility disturbances (Pri=Pi) in Eq. (2.10). 4. Conclusions Omitting indifference alternatives in a SC experiment can lead to loss of efficiency, as forcing individuals to opt for an alternative in an indifference situation may add white noise to the data, as has been previously established in the literature. Along the same line, allowing for indifference alternatives would offer a better depiction of individuals’ preferences and increase the richness of the dataset. Nevertheless, modelling with indifference alternatives is not trivial and the approaches reported in the literature only allow considering binary choice situations. We propose an alternative method that can be used in any choice situation. This approach exhibits high flexibility and allows specifying the likelihood function independent of the number of alternatives in the choice-set and in the subset of indifference alternatives. Finally, we test our method with help of four study cases. In the first two experiments (based on simulated and real data), we considered binary choices, observing that the proposed framework allows recovering the parameters used in the generation of the dataset without major difficulties outperforming other alternatives. In case studies 3 and 4 (also on the basis of simulated and real data) we considered further alternatives (specifically opting out). We observe that in most cases the proposed approach allows recovering the real underlying parameters (some problems may arise when considering large indifference thresholds and many indifference observations, but even in this case the proposed approach outperforms competing alternatives). Nevertheless, ignoring the extra variability associated with selecting an alternative as a part of an indifference set does not appear to have a significant effect on the estimated parameters, as is clearly shown in the third and fourth experiments. This way, it may be advisable for modellers to omit this feature, if the data has a scarce number of indifference observations. Further research should provide more evidence regarding this hypothesis. Alternative approaches, such as considering the indifference as an opt-out alternative or just ignoring the indifference preferences are clearly outperformed by the proposed framework and are not capable of recovering the real parameters. Nevertheless, datasets including a very small number of indifference observations do not appear to be severely affected by a poor treatment of indifference. In fact, completely ignoring these observations in case study four does not lead to significantly different results. Hence, the impact of indifference observations depends on their number. Finally, it can be concluded that individuals not stating a clear preference are indeed providing relevant information, which should be considered by the modellers: if you choose not to decide, you still have made a choice. Acknowledgments We wish to thank Ricardo Hurtubia for his useful suggestions and for pointing out the convenience of changing the name of the paper, suggesting we should choose a path that's clear, we should choose freewill. We are also grateful to Juan Pablo Sepúlveda for having provided us with the real data used in our first experiment. Finally, we are indebted to the Institute on Complex Engineering Systems (ICM: P-05-004-F; CONICYT: FB0816), the Centre for Sustainable Urban Development, CEDEUS (Conicyt/Fondap/ 15110020) and the Bus Rapid Transit Centre of Excellence funded by VREF (www.brt.cl), for their support. This article benefited greatly from the helpful comments of two anonymous referees.
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Appendix A See Fig. A1. In this case, respondents were instructed to choose “Any of them” if no alternative was favoured over the other.
Fig. A1. Presentation of choice situations in Case Study 2.
Appendix B See Fig. B1.
Fig. B1. Presentation of choice situations in Case Study 4.
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References Bradley, M.A., Daly, A.J., 1994. Use of the logit scaling approach to test for rank-order and fatigue effects in stated preference data. Transportation 21, 167–184. Bierlaire, M. 2003. BIOGEME: A free package for the estimation of discrete choice models, Proceedings of the 3rd Swiss Transportation Research Conference, Ascona, Switzerland Cantillo, V., Ortúzar, J. de D., 2006. Implications of thresholds in discrete choice modelling. Transp. Rev. 26, 667–691. Cantillo, V., Amaya, J., Ortúzar, J. de D., 2010. Thresholds and indifference in stated choice surveys. Transp. Res. Part B: Methodol. 44, 753–763. Carson, R.T., Louviere, J.J., Anderson, D.A., Arabie, P., Bunch, D.S., Hensher, D.A., Johnson, R.M., Kuhfeld, W.F., Steinberg, D., Swait, J., Timmermans, H., Wiley, J.B., 1994. Experimental analysis of choice. Mark. Lett. 5, 351–367. Chapman, R.G., Staelin, R., 1982. Exploiting rank ordered choice set data within the stochastic utility model. J. Mark. Res. 19, 288–301. Chorus, C., 2012a. Random regret minimization: an overview of model properties and empirical evidence. Transp. Rev. 32, 75–92. Chorus, C.G., 2012b. Logsums for utility-maximizers and regret-minimizers, and their relation with desirability and satisfaction. Transp. Res. Part A: Policy Pract. 46, 1003–1012. Coombs, C.H., Dawes, R.M., Tversky, A., 1970. Mathematical Psychology: An Elementary Introduction. Prentice-Hall, Englewood Cliffs, New Jersey. Dhar, R., 1997. Consumer preference for a no-choice option. J. Consum. Res. 24, 215–231. Dhar, R., Simonson, I., 2003. The effect of forced choice on choice. J. Mark. Res. 40, 146–160. Fenichel, E.P., Lupi, F., Hoehn, J.P., Kaplowitz, M.D., 2009. Split-sample tests of “no opinion” responses in an attribute-based choice model. Land Econ. 85, 348–362. Hess, S., Beck, M.J., Chorus, C.G., 2014. Contrasts between utility maximisation and regret minimisation in the presence of opt out alternatives. Transp. Res. Part A: Policy Pract. 66, 1–12. Kontoleon, A., Yabe, M., 2003. Assessing the impacts of alternative ‘opt-out’ formats in choice experiment studies: consumer preferences for genetically modified content and production information in food. J. Agric. Policy Resour. 5, 1–43. Krishnan, K.S., 1977. Incorporating thresholds of indifference in probabilistic choice models. Manag. Sci. 23, 1224–1233. Louviere, J.J., Hensher, D.A., Swait, J.D., 2000. Stated Choice Methods: Analysis and Applications. Cambridge University Press, Cambridge. McFadden, D., 1974. Conditional logit analysis of qualitative choice behaviour. In: Zarembka, P. (Ed.), Frontiers in Econometrics. Academic Press, New York, 105–142. Olsen, G.D., Swait, J.D., 1998. Nothing is Important. Working Paper, University of Calgary. Quandt, R.E., 1956. A probabilistic theory of consumer behaviour. Q. J. Econ. 70, 507–536. Roberts, J.H., Lattin, J.M., 1991. Development and testing of a model of consideration set composition. J. Mark. Res. 28, 429–440. Schlereth, C., Skiera, B., 2016. Two new features in discrete choice experiments to improve willingness-to-pay estimation that result in SDR and SADR: separated (adaptive) dual response. Manag. Sci.. http://dx.doi.org/10.1287/mnsc.2015.2367. Swait, J.D., Erdem, T., 2007. Brand effects on choice and choice set formation under uncertainty. Mark. Sci. 26, 679–697. Thurstone, L.L., 1927. A law of comparative judgment. Psychol. Rev. 34, 273–286.
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Contents lists available at ScienceDirect
Journal of Choice Modelling journal homepage: www.elsevier.com/locate/jocm
If you choose not to decide, you still have made a choice Francisco J. Bahamonde-Birke
a,b,c,⁎
d
, Isidora Navarro , Juan de Dios Ortúzar
MARK e
a
Institut für Verkehrsforschung, Deutsches Zentrum für Luft- und Raumfahrt (DLR), Germany b Energy, Transportation and Environment Department, Deutsches Institut für Wirtschaftsforschung, Berlin, Germany c Technische Universität Berlin, Germany d Department of Transport Engineering and Logistics, Pontificia Universidad Católica de Chile, Chile e Department of Transport Engineering and Logistics, Centre for Sustainable Urban Development (CEDEUS), Pontificia Universidad Católica de Chile, Chile
A R T I C L E I N F O
ABSTRACT
Keywords: Discrete choice models Indifference Stated-choice experiments
When designing stated-choice experiments modellers may consider offering respondents an “indifference” alternative to avoid stochastic choices when utility differences between alternatives are perceived as too small. By doing this, the modeller avoids adding white noise to the data and may gain additional information. This paper proposes a framework to model discrete choices in the presence of indifference alternatives. The approach allows depicting the likelihood function, independent of the number of alternatives in the choice-set and in the subset of indifference alternatives, offering a new approach to existing methods that are only defined for binary choice situations. The method is tested with the help of simulated and real data observing that the proposed framework allows recovering the parameters used in the generation of the synthetic datasets without major difficulties in most cases. Alternative approaches, such as considering the indifference option as an opt-out alternative or ignoring the indifference choices are clearly outperformed by the proposed framework and appear not capable of recovering parameters in the simulated set.
1. Introduction Discrete choice models rely on the assumption that individuals are rational decision makers that maximize their utility when facing a given choice situation. This way, individuals will opt for a given alternative if and only if it promises them the maximum expected utility among all alternatives in their choice-sets (Thurstone, 1927; McFadden, 1974). Nevertheless, establishing which alternatives should be considered into the individuals’ choice-set is not an easy task. In real situations the modeller will just observe the chosen alternatives and needs to construct the choice-sets of the individuals on the basis of their characteristic and their choices. This involves major difficulties, as it is well established that people tend to narrow their decisions to only a subset of the potentially available options (Roberts and Lattin, 1991; Swait and Erdem, 2007). By contrast, when dealing with stated preference (SP) data the choice-set must be established a priori. In this case, it is important that it be carefully defined preserving the realism of the choice situations. Thus, in many cases it might be necessary to consider an opt-out (non-purchase) alternative (Carson et al., 1994; Olsen and Swait, 1998); whether this alternative should be included directly into the choice-set (Louviere et al., 2000) or indirectly via dual response procedures (Dhar and Simonson, 2003) remains a debatable point (see Schlereth and Skiera (2016) for a good discussion), but the necessity of alternatives accounting for a reservation utility level
⁎
Corresponding author at: Institut für Verkehrsforschung, Deutsches Zentrum für Luft- und Raumfahrt (DLR), Germany. E-mail addresses: [email protected], [email protected] (F.J. Bahamonde-Birke), [email protected] (I. Navarro), [email protected] (J.d.D. Ortúzar). http://dx.doi.org/10.1016/j.jocm.2016.11.002 Received 23 February 2016; Received in revised form 23 November 2016; Accepted 26 November 2016 1755-5345/ © 2016 Published by Elsevier Ltd.
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higher than the expected utility of all alternatives in the choice-set is well-established (Kontoleon and Yabe, 2003). A similar but far less analysed problem is the inclusion of indifference alternatives in the choice-set. If the modeller does not allow for respondents to state their indifference among two or more alternatives, they will be forced to opt for one of them in a rather stochastic manner, adding white noise to the experiment. Additionally, doing so would provide less information about the individuals’ preferences leading to loss of efficiency. Indeed, Cantillo et al. (2010) used a synthetic dataset to show that assigning the preferences associated with an indifference alternative randomly, diminished significantly the model's capability to recover the input parameters. Furthermore, Cantillo et al. (2010) also considered real databanks, observing than offering the possibility of stating indifference may indeed affect the outcome of the experiment (the estimated parameter values). Along these lines, empirical evidence (Dhar, 1997; Fenichel et al., 2009) shows that in experiments including non-purchase options, indifference situations may artificially increase the probability of selecting the opt-out1 alternative, as a kind of cognitive bias. Therefore, it appears advisable to include indifference alternatives when also offering non-purchase options as a way to reduce cognitive biases. Nevertheless, including indifference alternatives should be carefully considered, as it might generate other kind of complications, especially if individuals are overwhelmed by the complexity of the choice experiment. Both situations (opt-out and indifference alternatives) exhibit, however, substantial differences; while the former suggest the existence of a reservation utility that is higher than the utility provided by the alternatives in the choice set, the latter indicates that individuals ascribe the same utility to two or more alternatives in it (this utility being higher than the reservation utility). Therefore, in the first case an extra alternative accounting for this reservation prize should be considered. Nevertheless, considering an extra alternative to reflect indifference choices does not seem to be appropriate, as it does not reflect the causes leading to the statement of indifference; in fact, by treating indifference as a new (opt-out) alternative, the modeller implicitly assumes that the utility ascribed to this new option would be greater than that of the competing alternatives, which is clearly not the case. Despite the fact, that according to classical theory indifference situations will only arise if the expected utility of two or more alternatives is the same (curves of indifference), the underlying behavioural theory behind the indifference phenomena suggests the existence of perception thresholds, below which the individuals are not able to perceive differences between two stimuli (Quandt, 1956; Coombs et al., 1970; Cantillo and Ortúzar, 2006). Krishnan (1977) developed an operational discrete choice model accounting for the existence of indifference thresholds. This approach (Minimum Perceivable Differences model, MPD) allows taking into account the fact that observations falling into the indifference interval would be assigned stochastically to one alternative, in the context of a binary choice situation. Cantillo et al. (2010) expanded the MPD-approach to allow for individuals stating their indifference in stated-choice (SC) experiments. This way, the indifference alternative would be selected if the difference between the utility of both alternatives was smaller than a threshold, to be estimated. The main limitation of the MPD-approach is that it only allows considering binary choice situations. Thus, the method can neither consider situations where two alternatives exhibit a similar utility (which is superior to all other alternatives in the choiceset), nor cases when three or more choices report an apparently identical utility (which may be of particular interest when considering alternatives to first-choice SP experiments, such as rankings). The same limitation arises, when considering approaches such as an ordered logit framework (with indifference being an intermediate choice between two binary alternatives). A method that allows accommodating more than two alternatives, consists in assuming that instead of behaving as utility maximizers, the individuals minimize their regret (RRM framework; Chorus, 2012a). Under this assumption, the regret associated with a certain alternative is given by the direct comparison of its attributes with those of all remaining alternatives in the choice set (whereby only a negative performance would generate regret). Thus, including an extra null-alternative (without attributes) into the model cannot longer be associated with a higher reservation utility, but would rather stand for a level of regret, above which none of the alternatives in the choice-set is favoured (i.e. none of the alternatives significantly minimizes the regret in comparison with other options in the choice-set, Chorus, 2012b). Hence, under this assumption an extra alternative would allow to capture indifference (Hess et al., 2014). Nevertheless, this approach does not appear appropriate to deal with non-binary choice situations, as the extra alternative would be indicative of indifference among the whole choice-set, not allowing to consider indifference among a sub-set of alternatives (e.g. the respondent is indifferent among two alternatives, but both alternatives are preferred over the remaining alternatives in the choice-set). Furthermore, it does not seem appropriate to consider experiments allowing for both the option of stating indifference as well as opting out, because: (i) it would be necessarily a non-binary situation, and (ii) it would imply considering two different nullalternatives, which should account for two completely different phenomena. Finally, the approach would necessarily require the analyst to assume a regret minimization strategy (ideally, the modeller should aim at an approach that allows considering indifference under different assumptions, e.g. regret minimization or utility maximization, and discern between them on an empirical basis). This paper discusses the implications of indifference choices concerning the utility ascribed to the different alternatives in the choice-set. Along these lines, the paper presents a new approach that allows dealing with indifference choices in multinomial choice situations. This framework allows not only addressing first-choice SP experiments but also rankings, where indifference choices may be expected to appear more often. The approach is tested with the help of simulated and real datasets, observing that it clearly
1 For the purposes of this paper, it is assumed that opting-out implies that none of the alternatives satisfies the individual's requirements (i.e. a non-purchase option). This is highly recommended in non-pivoted SP experiments, as with totally new options it could well be that none is acceptable and not presenting an opt-out may bias results (Olsen and Swait, 1998).
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outperforms other methods allowing for a proper estimation of parameters. 2. Modelling with indifference alternatives According to random utility theory (Thurstone, 1927; McFadden, 1974) in a given choice situation an individual q would opt for alternative i belonging to choice-set A if and only if Uiq > Ujq ∀ j ≠ i ∈ A. Then, the probability with which the individual would select this alternative is given by:
Piq = P(Uiq > Ujq )
∀j≠i∈A
(2.1)
The expected utility U can be expressed in terms of a representative component V, characterized through concrete and measurable properties of the alternatives and the individuals, and an error term ε representing all unknown (for the analyst) elements of the decision. Assuming an additive specification (other structures are also possible, but for didactic purposes we assume the more common specification), (2.1) can be rewritten as:
Piq = P(Viq + εiq > Vjq + εjq ) = P(Viq − Vjq > εjq − εiq )
∀j≠i∈A
(2.2)
Finally, the likelihood of observing a set of given choices is given by:
L=
∏ ∏ Pjq yjq, q
(2.3)
j∈A
where yjq takes the value of one if alternative j is selected by individual q and zero otherwise. 2.1. Stating indifference If the modeller allows for individual q to state his indifference between n alternatives belonging to an indifference-set B, which is a subset of the complete choice-set A, (consisting of m alternatives, with m≥n), the choice probability of the alternatives belonging to B, may be written as follows:
Pi = Pj = Pk = …
∀ i, j, k , … ∈ B
(2.4)
As under random utility theory all alternatives belonging to B must maximize the individual's expected utility U, Eq. (2.4) can be rewritten in these terms:
Ui ≈ Uj ≈ Uk ≈ …
∀ i, j, k , … ∈ B,
(2.5)
where the inequality accounts for the existence of utility differences (smaller than a given threshold) that are not being perceived by the respondents. If we introduce error terms ϕ, then Eq. (2.5) can be expressed in terms of equalities (where ϕ is an error term merely guaranteeing that the utility differences within a given indifference-set be equal to zero):
Ui + ϕi = Uj + ϕj = Uk + ϕk = …
∀ i, j, k , … ∈ B
(2.6)
Then, expressing the expected utility in terms of a representative utility V and the aforementioned error terms ε (accounting for all unknown elements of the decision but the indifference thresholds leading to the statement of indifference) leads to the following expression:
Vi + εi + ϕi = Vj + εj + ϕj = Vk + εk + ϕk = …
∀ i, j, k , … ∈ B
(2.7)
Finally, the probability associated with an alternative in the indifference set would be given by:
Pri = P(Vi − Vj > εj + ϕj − εi − ϕi )
∀ j ≠ i ∈ A ∧ ϕj = 0, ∀ j ∉ B
(2.8)
where Pri≠Pi, as it is associated with a larger error component (due to ϕ). When modelling with indifference alternatives, it cannot be assumed that all alternatives in the indifference-set are selected at the same time, but rather that all are selected with a frequency 1/n. Hence, when considering indifference options, the likelihood of observing a given choice can be expressed as:
L=
∏ Prj yj, (2.9)
j∈A
where yj takes the value of 1/n if alternative j∈B and zero otherwise. If we consider a group of individuals M selecting indifferencesets and another group of individuals L selecting unique preferred alternatives (where M and L are mutually exclusive), the general likelihood function for the population will take the following form:
L=
∏ ∏ Pjq yj⋅ ∏ ∏ Prjq yj, q∈M j∈A
(2.10)
q∈L j∈A
It is important to notice that using a value of yj equal to one for all alternatives in an indifference-set would result in overweighting all observations by respondents selecting indifference options. Also, note that the likelihood function would be maximized when all alternatives in an indifference-set are equally likely (AM-GM inequality). Even though, the above framework was derived for utility maximization, it is straightforward to extend it to consider regret 15
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minimization strategies (Chorus, 2012a). Furthermore, as the meaning of null-alternatives is different when pursuing regret minimization and utility maximization strategies, this approach would allow for a fair comparison between both sets of assumptions in the presence of indifference. This would allow for the modeller to discern which set of assumptions offers a better representation of the underlying decision-making process on the basis of the empirical evidence. Similarly, it is easy to extend it for ranking preference elicitation procedures when two or more alternatives at different depths of the ranking order are considered equivalent (i.e. the respondent cannot choose among them). In this case, the modeller would rely on the choices arising by exploiting the ranking2 (Chapman and Staelin, 1982; Bradley and Daly, 1994) and the framework would apply to those cases where there is indifference between two or more alternatives (i.e. allowing to deal with ties at any depth of the ranking order). 2.2. Identification of the error term ϕ Identification and the distribution of the error terms ϕ are complicated topics. First, the errors only appear in the utility function if the alternatives are selected as part of an indifference-set (as part or Pri), which may significantly decrease the number of observations available for estimation. Furthermore, this additional error term will only be associated with the alternatives that are indeed part of a given indifference set; thus, the utility functions of the remaining alternatives (not selected as a part of the indifference set) will not be affected by this term. For this reason, non-symmetrical assumptions regarding its distribution are unsuitable (the estimator would diverge, as it would only be associated with selected alternatives). Moreover, establishing an adequate functional form for the error term ϕ is not an easy task, as it does not comply with the usual assumptions regarding error terms. In fact, ϕ is an error component that merely guarantees that the difference among two or more different utilities is equal to zero; that is, the error is being added to the alternatives associated with the lesser utilities or subtracted from the alternatives associated with higher utilities, so that the utility of all alternatives in the indifference-set add up to the same amount. Hence, these errors would not be properly represented by the usual assumptions concerning error terms. That, in conjunction with the aforementioned unsuitability of non-symmetrical distributions, creates major difficulties. If the modeller assumes that the error term ϕ follows a distribution equal to the difference between two Logistic distributions with different scale parameters,3 the sum εi − εj + ϕ would also be represented by a Logistic distribution, with a smaller scale parameter (i.e. with a larger standard deviation). As a consequence, the utility functions of alternatives being selected as a part of an indifference-set would be equivalent to the utility functions of the single-choice framework (Pi), but would be associated with a smaller scale parameter, reflecting the increased uncertainty.4 A limitation of this approach is that it does not allow quantifying the thresholds leading to the statement of indifference; notwithstanding, the model stills offers an adequate depiction of the underlying utility functions associated with the different alternatives as well as a functional form for forecasting. Additionally, the approach allows considering different scale parameters depending on the number of alternatives in the indifference-sets, or on which alternatives are being considered as part of the indifference-sets, as well as on the socio-economic characteristics on the individuals. Because of identification issues, the modeller is forced to normalize one of the aforementioned scale parameters (either that related to the equations for single choices – when the respondent selects a unique alternative - or that associated with alternatives selected as part of an indifference-set). Both normalizations would lead, however, to different estimates; if the modeller normalizes the error associated with the single choices, s/he would be scaling the estimates upwards (when comparing them with the outcomes of an alternative model, where the scale parameter was fixed at one for the whole sample), as single choices are related to a lesser degree of uncertainty (i.e. the differences of representative utilities are larger, in average, when indifference situations are left out or alternatively, indifference situations would arise more often when utility differences are smaller). Thus, when fixing the scale parameter for single choices the estimates are no longer comparable with alternative specifications.5 In this case, the analyst does not attempt to estimate separate models for individuals choosing single alternatives or indifference sets, but rather to estimate a parsimonious model that is consistent with both kind of choices (and the estimates of which are comparable with models estimated without introducing the error term ϕ). A possible way to deal with this problem is to fix the scale parameter associated with the weighted average of the sample (normally at one, but other values may be chosen). In this form, the analyst would estimate k-1 parameters (with k representing the number of different scale parameters), and express the last one in terms of the others to be estimated as in:
λ⋅N = λ1⋅n1 + λ 2⋅n 2 + λ 3⋅n3 + …,
(2.11)
where N represents the total number of observations in the sample, λ the weighted average of the scale parameters, λj scale parameters to be estimated, and nj the number of observations associated with a given scale parameter. A final characteristic of the approach is that when considering only two alternatives, different scale parameters cannot be 2 By exploiting the ranking the analyst assumes that the first alternative (in the ranking) would be selected in a choice-set considering all alternatives. The second ranked alternative would be the choice in a set including all alternatives but the first ranked, and so on. 3 While this assumption does not appear to be wholly appropriate to describe the aforementioned response, it evidences itself as fairly convenient for modelling issues. 4 This framework resembles the structure used to deal with RP and SP data simultaneously. 5 Naturally, as the scale parameters are fixed without loss of generality, the model would still be appropriate, but a correction would be required in order to make different specifications comparable.
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estimated. This relates to the fact that when all alternatives are equally probable, all estimates must necessarily be equal to zero and therefore any scale parameter multiplying a group of indifference sets containing all alternatives would be unidentified (and therefore can be excluded without loss of generality). As in binary choice situations every indifference-set must necessarily include both alternatives, considering different scale parameters in this case would be redundant. As a corollary, considering different scale parameters would only impact the model when those can be associated with at least one indifference-set (selected by at least a part of the population) not containing all alternatives in the choice-set. 3. Study cases To test our framework as well as the impact of introducing indifference alternatives into the model, we conducted four different empirical tests. First, we considered a simulation exercise consisting of only two alternatives (binary choice situation). Then, we analysed a real case considering a similar binary structure. Then, a second simulation exercise introduced an opt-out alternative (additional to the possibility of stating indifference), and finally, we tested the framework using real data consisting of a binary choice set allowing for indifference as well as for opting out. 3.1. Simulated binary choices Our first simulated scenario considered binary choice situations where the utility functions of the alternatives were given by:
U1 = β1⋅X1 + ε1 U2 = ASC2 + β2⋅X2 + ε2
(3.1)
Here, individuals are supposed to choose the alternative with the highest expected utility or to state their indifference if the utility difference between both alternatives is smaller than a given threshold. We took X1 and X2 as random draws from Normal distributions with mean 0 and −1, and standard deviations 1 and 1.5, respectively. The error terms were assumed to be independently EV1 distributed with mean 0 and scale parameter 1; both β parameters as well as the ASC were fixed to 1, while three different values were tested for the threshold: 0.15, 0.3 and 0.45 representing small, medium and large indifference intervals, respectively. In all three cases 5000 pseudo-individuals were generated observing that 254, 495 and 739 stated their indifference between both alternatives, respectively. With this dataset we estimated four models, considering different approaches to address the indifference alternatives. Model 1 considered all indifference options as an extra alternative; thus, in our utility maximization framework this extra alternative represents opting-out rather than indifference.6 Model 2 presents the results for a case where the individuals stating their indifference are ignored. Model 3 does not account for the fact that all alternatives in a selected indifference-set are not observed with the same frequency (if a given alternative is selected as a part of a binary indifference set, in reality, it would only be selected with a frequency of 0.5) as alternatives being selected directly; hence, Model 3 assigns a weight (yj) of 1 to all observations. Finally, Model 4 considered the proposed framework (Eq. (2.10)). All models were estimated using PythonBiogeme (Bierlaire, 2003), and the results are presented in Table 3.1 (the standard deviation of the estimated parameters is shown in parenthesis). As can be observed, considering the indifference options as opt-out alternatives yields a significantly worse goodness-of-fit than the other approaches. Nevertheless, in this particular case the log-likelihood does not offer meaningful insights, as Model 1 considers one alternative more (affecting the choice probabilities), while Models 2 and 3 consider less/more observations (ignoring/ overweighting the observations associated with smaller utility differences would artificially diminish/increase the error and goodness-of-fit). An analysis of the parameters provides more useful information. Models 1–3 are not capable of recovering the values used in the generation of the dataset (the t-tests of equality are rejected at a significance level of 5%, for at least part of the estimators in every case). This may be related to the fact that the artificial increase/descent of the error affects the scale parameter, biasing the results. Model 4 (the proposed framework) recovers the parameters used in the generation of the dataset without major difficulties. Regarding the indifference intervals, it can be observed that the differences between the approaches increase (as well as the difficulties associated with the parameter recovery for Models 1–3) as the indifference threshold gets greater leading to more individuals stating their indifference. Model 4 performs adequately in every case. 3.2. Real dataset: binary choices The second study case consists of three different datasets that were collected as a part of the same survey on passenger rail transportation in Southern Chile (Appendix A presents an example of the way alternatives were presented to the individuals). The different datasets are associated with three different kinds of trips: short and long interurban trips and urban trips. In this regard, the individuals were asked to choose between two unlabelled hypothetical alternatives, having also the choice of stating their indifference among both alternatives. All experiments considered the same attributes: fare (P), travel time (TT), access time (AT) and 6 As we are considering a binary choice situation in this specific case, it would have been possible to test the approach put forward by Hess et al. (2014) by assuming regret minimization. Nevertheless, such comparison would be spurious as we assumed utility maximization when constructing the simulated dataset; the same applies (though to a lesser extent) to alternatives methods, such as the MPD-approach (Cantillo et al., 2010).
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Table 3.1 Estimated models. Case study 1. Threshold
Variable
Model 1
Model 2
Model 3
Model 4
0.15
X1 X2 ASC2 ASCopt-out
0.970 (0.0392) 0.964 (0.0307) 0.905 (0.0454) −2.30 (0.0679)
1.10 (0.0445) 1.07 (0.0345) 1.04 (0.0503) –
0.973 (0.0397) 0.950 (0.0304) 0.909 (0.045) –
1.03 (0.0419) 1.01 (0.0323) 0.968 (0.0474) –
Final log-likelihood No. of observations
−3248.472 5000
−2190.648 4746
−2626.479 5254
- 2410.844 5000
X1 X2 ASC2 ASCopt-out
0.897 (0.0379) 0.969 (0.0304) 0.933 (0.0451) −1.55 (0.0516)
1.14 (0.0430) 1.16 (0.0344) 1.19 (0.0489) –
0.904 (0.0381) 0.933 (0.0294) 0.934 (0.0438) –
1.00 (0.0401) 1.03 (0.0314) 1.04 (0.0459) –
Final log-likelihood No. of observations
−3693.051 5000
−1978.882 4505
−2812.637 5495
- 2403.098 5000
X1 X2 ASC2 ASCopt-out
0.846 (0.0366) 0.906 (0.0294) 0.872 (0.0441) −1.05 (0.0445)
1.15 (0.0484) 1.15 (0.0383) 1.18 (0.0561) –
0.819 (0.0353) 0.811 (0.0267) 0.834 (0.0413) –
0.945 (0.0402) 0.941 (0.0309) 0.965 (0.0468) –
Final log-likelihood No. of observations
−4082.823 5000
−1868.977 4261
−3109.345 5739
- 2505.769 5000
0.3
0.45
transport mode (TM – indicating if the trip was made by train or by bus). We considered the following utility functions:
U1 = ASC1 + βP ⋅P1 + βTT ⋅TT1 + βAT ⋅AT1 + βTM ⋅TM1 + ε1 U2 = ASC2 + βP ⋅P2 + βTT ⋅TT2 + βAT ⋅AT2 + βTM ⋅TM2 + ε2
(3.2)
The first dataset (short interurban trips) consists of 2781 observations, of which 289 (10.4%) stated indifference among both options. The second dataset (long interurban trips) includes 2745 observations, with 202 (7.4%) indicating indifference. Finally, the data on urban trips consists of 5841 observations, 553 (9.5%) of which were associated with indifference. For consistency purposes we considered basically the same models as in the previous study case. This way, Model 1 considers indifference as an additional opt-out alternative, Model 2 ignores all indifference observations, Model 3 relays on the proposed approach without accounting for frequency considerations (in fact, duplicating the weight of the indifference observations) and Model 4 follows the proposed approach, as depicted in Eq. (2.10). For illustrative purposes, this time we also included a model considering a null-alternative in the context of a regret minimization framework (Model 5). In this case, the additional alternative, can be framed as indifference (Hess et al., 2014).7 The results are presented in Table 3.2. In line with the previous study case, the results confirm that treating indifference as an opt-out alternative (in the utility maximization framework) lead to results that widely diverge from alternative approaches (however, it is not possible to establish positively that the results are definitely biased, as it is not possible to observe the underlying model). Along these lines, as in the previous case, the estimates of Model 2 and Model 3 are consistently deviated up and down, respectively, from the results associated with Model 4. All three considered cases (interurban short and long trips, as well as urban trips) exhibit exactly the same patterns. Finally, and according with the expectations, the estimates associated with Model 5 are similar, but slightly larger, than those obtained for Model 4 (with the exception of the ASCs, as they are indicative for utility in RUM models and regret in RRM models). However, a direct comparison between both models cannot be performed, as they consider a different number of alternatives (although, surprisingly, Model 5 exhibits a worse goodness-of-fit than Model 1 for urban trips). These results confirm our previous findings, implying that treating indifference as an opt-out alternative, as well as ignoring indifference observations or applying the proposed approach without considering frequency issues may lead to biased results. This evidence supports the necessity of an adequate modelling approach when dealing with indifference observations and real data. 3.3. Simulated binary choices including an opt-out alternative A second simulated scenario was considered. The dataset was generated similarly to the previous one, but in this case we introduced an opt-out alternative,8 as shown in the following equations set. 7 It could have also been possible to consider Models 2–4 on the basis of regret minimization, but given that they are binary cases, regret minimization and utility maximization would lead to the same outcome. 8 Even though we have framed the third alternative as opting-out, it is easy to see that, for the purposes of the modelling, it would be equivalent to considering a choice-set with three alternatives.
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Table 3.2 Estimated models. Case study 2. Survey
Variable
Model 1
Model 2
Model 3
Model 4
Model 5
Short Trips
P TT AT TM ASC2 ASCopt-out
−0.83 (0.204) −0.172 (0.0371) −0.067 (0.0546) 0.148 (0.0788) −0.493 (0.0779) −4.06 (0.45)
−1.99 (0.343) −0.375 (0.0662) −0.0778 (0.0673) 0.272 (0.0878) −0.507 (0.128) –
−1.63 (0.301) −0.299 (0.0596) −0.0429 (0.0592) 0.202 (0.0781) −0.402 (0.114) –
−1.79 (0.32) −0.332 (0.0626) −0.0571 (0.0628) 0.232 (0.0825) −0.447 (0.12) –
−1.97 (0.337) −0.351 (0.0635) −0.0667 (0.0684) 0.258 (0.0869) 0.482 (0.125) 4.5 (0.0879)
log-likelihood No. of obs.
2781.00 2781
−1620.24 2492
−2041.62 3070
−1832.10 2781
− 2549.21 2781
P TT AT TM ASC2 ASCopt-out
−0.855 (0.161) −0.112 (0.0342) −0.112 (0.0534) 0.232 (0.0792) 0.475 (0.0964) −6.17 (0.81)
−1.06 (0.202) −0.344 (0.0508) −0.123 (0.066) 0.268 (0.0876) 0.0866 (0.137) –
−0.876 (0.185) −0.278 (0.0465) −0.102 (0.0601) 0.264 (0.0812) 0.0769 (0.126) –
−0.957 (0.192) −0.308 (0.0485) −0.111 (0.0628) 0.267 (0.0841) 0.0807 (0.131) –
−0.988 (0.194) −0.352 (0.0513) −0.126 (0.0668) 0.28 (0.0866) −0.0428 (0.137) 4.64 (0.108)
log-likelihood No. of obs.
−2403.90 2745
−1666.02 2543
−1960.91 2947
−1814.13 2745
−2387.72 2745
P TT AT TM ASC2 ASCopt-out
−1.1 (0.263) −0.452 (0.0432) −0.169 (0.0659) 0.3 (0.052) −0.318 (0.0733) −3.38 (0.187)
−1.25 (0.273) −0.427 (0.0455) −0.178 (0.0689) 0.277 (0.059) −0.257 (0.0763) –
−1.01 (0.246) −0.353 (0.0406) −0.146 (0.0616) 0.236 (0.0533) −0.217 (0.068) –
−1.12 (0.258) −0.386 (0.0428) −0.16 (0.0649) 0.255 (0.0559) −0.235 (0.0717) –
−1.2 (0.268) −0.417 (0.0443) −0.172 (0.0673) 0.284 (0.059) 0.261 (0.0745) 4.49 (0.0582)
log-likelihood No. of obs.
−5408.35 5841
−3589.52 5288
−4368.88 6394
−3979.80 5841
−5419.95 5841
Long Trips
Urban Trips
U1 = β1⋅X1 + ε1 U2 = ASC2 + β2⋅X2 + ε2 Uopt−out = ASCopt − out + ε3
(3.3)
The data generation process was analogous to the previous section, but this time the pseudo-individuals were allowed to state their indifference among: (i) both alternatives, (ii) one existing alternative and opting-out, and (iii) both existing alternatives and opting-out. The ASC associated with the opt-out alternative (accounting for the reservation utility) was fixed at zero. Again we considered the same different indifference thresholds. In the first case (low threshold), we observed 400 observations indicating indifference (124 between both existing alternatives, 131 between alternative 1 and opting out, 135 between alternative 2 and opting out, and 10 between both alternatives and opting out). In the second case (intermediate indifference threshold) 745 pseudo-individuals stated their indifference (200 between both existing alternatives, 260 between alternative 1 and opting out, 230 between alternative 2 and opting out, and 55 between both alternatives and opting out), while in the third case (large indifference threshold) we observed 1093 indifference observations (297 between both existing alternatives, 345 between alternative 1 and opting out, 319 between alternative 2 and opting out, and 132 between both alternatives and opting out). In this case, considering extra alternatives does not make sense, as option 3 already accounts for the opt-out alternative. This way, we estimated four different models. Model 1 presents the results of ignoring the indifference choices. Model 2 considers the proposed framework but ignores the error term ϕ and does not consider a different scale parameter for the observations associated with indifference-sets. Model 3 considers different scale parameters but assigns a weight (yj) of 1 to all observations. Finally, Model 4 represents the proposed framework. The scale parameters were fixed in accordance with (2.10) and (2.11). The results are presented in Table 3.3. This time, all specifications but Model 1 (which ignores the indifference observations) allow for an adequate estimation of the parameters used in the generation of the dataset. Model 2 performs surprisingly well, but still offers a significantly worse goodnessof-fit than Model 4. Nevertheless, the fact that Model 2 performs acceptably while ignoring the different variability, suggest that considering different scale parameters may be omitted without major complications if the modeller does not count with enough indifference-set observations (to allow for a correct estimation of the λ parameter). Model 3 performs significantly better than its un-weighted counterpart in the previous simulation exercise. The reason is that when considering different scale parameters, the weighting only affects the observations multiplied by the same scale parameter. Then, in this case, the weights associated with the single choices are not relevant, while we observed a small amount of observations with a weight of 1/3 (individuals stating their indifference among all choices) and a large majority associated with a weight of 1/2, which clearly diminishes the bias associated with ignoring the weighting. In fact, if we ignore the scale parameter (analogously to 19
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Table 3.3 Estimated models. Case study 3. Threshold
Variable
Model 1
Model 2
Model 3
Model 4
0.15
X1 X2 ASC2 ASCopt-out λIndifference-set
1.09 (0.0430) 1.05 (0.0334) 1.11 (0.0493) 0.0242 (0.0419) –
1.03 (0.0404) 0.991 (0.0310) 1.05 (0.0464) 0.0272 (0.0396) –
1.04 (0.0401) 0.999 (0.0310) 1.05 (0.0461) 0.0283 (0.0391) 0.436 (0.0511)
1.04 (0.0406) 0.999 (0.0314) 1.05 (0.0466) 0.0257 (0.0395) 0.439 (0.0717)
Final log-likelihood No. of observations
3848.233 4600
−4294.822 5000
−4699.123 5410
4268.096 5000
X1 X2 ASC2 ASCopt-out λIndifference-set
1.08 (0.0441) 1.08 (0.036) 1.08 (0.0514) 0.0394 (0.0431) –
0.976 (0.0391) 0.963 (0.0311) 0.996 (0.0461) 0.0304 (0.0388) –
0.979 (0.0383) 0.968 (0.0309) 0.989 (0.0449) 0.0302 (0.0377) 0.438 (0.0385)
0.98 (0.0393) 0.974 (0.0318) 0.981 (0.046) 0.0355 (0.0383) 0.447 (0.0535)
Final log-likelihood No. of observations
3572.903 4255
−4404.245 5000
−5202.828 5800
−4356.509 5000
X1 X2 ASC2 ASCopt-out λIndifference-set
1.17 (0.0468) 1.13 (0.0383) 1.19 (0.0559) 0.0242 (0.0419) –
0.979 (0.0386) 0.776 (0.0272) 1.00 (0.0452) −0.0785 (0.0398) –
0.937 (0.0383) 0.858 (0.0291) 0.955 (0.0448) 0.0291 (0.0366) 0.153 (0.0355)
0.933 (0.0390) 0.872 (0.0298) 0.941 (0.0456) 0.0267 (0.0362) 0.142 (0.0479)
Final log-likelihood No. of observations
−3163.748 3907
−4515.938 5000
−5701.262 6225
−4360.298 5000
0.3
0.45
Model 2 but ignoring the weighting), the model is no longer capable of recovering the parameters. Regarding the magnitude of the threshold we observe that all specifications (besides Model 1) perform adequately as long as the indifference threshold is small or intermediate. When the indifference threshold is large (causing that an important part of our pseudo-population is associated with indifference alternatives), all models exhibit some difficulties recovering the parameters. This may be explained by the fact that a larger indifference threshold may be indeed associated with a larger uncertainty and that our assumptions regarding the distribution of the error term ϕ are not exactly accurate. Even in this case, Model 4 exhibits the best performance, and is capable of recovering three out of four parameters and clearly outperforms Model 1 and Model 2. 3.4. Real dataset including opt-out and indifference options For the fourth case study we used another real dataset. Information was collected as part of a project aiming to identify and assess the value of urban attributes (Appendix B presents an example of the way alternatives were presented to the individuals). For this purpose, a SP experiment in the context of residential location choice was designed and conducted in one neighbourhood of Santiago de Chile. The area has some special characteristics, such as high income, urban amenities and enhanced possibilities (in terms of transport facilities). The attributes considered in the experiment were the existence of Green Areas on the sidewalk (GA), Bus Corridors (BC), Green Verges segregating the bus corridors (GV), Bike Lanes (BL), and changes in the housing rent (ΔP). The experiment offered respondents four different options; two alternatives depicted through visual representations (with utilities U1 and U2), the possibility of stating indifference between both of them, as well as an opt-out alternative (with utility Uopt-out), as in the following equations:
U1 = ASC1 + βga⋅GA1 + βgv⋅GV1 + βbike⋅BL1 + βbus⋅BC1 + βprice⋅ΔP1 + ε1 U2 = ASC2 + βga⋅GA2 + βgv⋅GV2 + βbike⋅BL 2 + βbus⋅BC2 + βprice⋅ΔP2 + ε2 Uopt−out = ASCopt − out + ε3
(3.4)
The dataset has 588 observations, twelve of which (2.04%) are associated with indifference statements, and 26 with the opt-out alternative. We estimated four different models (Table 3.4) similar to those described in the third case study. Model 1 ignores the individuals stating their indifference. Model 2 does not consider different scale parameters for the observations associated with the indifference sets, while Model 3 adds different scale parameters but considers the same weight for all observations; finally, Model 4 uses the proposed framework (different scale parameters and different weights). As can be observed, in this case all estimated models perform similarly; in fact, it is not possible to identify statistically significant differences between the parameters estimated by each model. This may be related to the fact that only twelve individuals (2.04%) stated their indifference among alternatives. It implies that considering involved specifications is not crucial when the number of 20
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Table 3.4 Estimated models. Case study 4. Variable
Model 1
Model 2
Model 3
Model 4
GA GV BL BC ΔP ASC1 ASC2 λindifference-set
0.204 (0,112) 0,392 (0.138) 0.977 (0.120) −0.133 (0.130) −1.97 (0.407) 1.60 (0.235) 1.54 (0.236) –
0.196 (0.110) 0.392 (0.136) 0.956 (0.118) −0.129 (0.128) −1.93 (0.400) 1.64 (0.233) 1.58 (0.235) –
0.189 (0.109) 0.391 (0.134) 0.937 (0.118) −0.126 (0.127) −1.89 (0.394) 1.68 (0.232) 1.62 (0.234) 0.977 (0.344)
0.196 (0,110) 0,391 (0.136) 0.957 (0.119) −0.129 (0.128) −1.93 (0.400) 1.64 (0.234) 1.58 (0.235) 0.967 (0.474)
Final log-likelihood No. of observations
−438.742 576
−448.481 588
−458.165 600
−448.478 588
indifference observations is relatively low. In line with the previous case study it is not possible to identify differences associated with the inclusion of different scale parameters for the indifferent sets, which supports the hypothesis that the more involved approach does not perform substantially better than assuming equal utility disturbances (Pri=Pi) in Eq. (2.10). 4. Conclusions Omitting indifference alternatives in a SC experiment can lead to loss of efficiency, as forcing individuals to opt for an alternative in an indifference situation may add white noise to the data, as has been previously established in the literature. Along the same line, allowing for indifference alternatives would offer a better depiction of individuals’ preferences and increase the richness of the dataset. Nevertheless, modelling with indifference alternatives is not trivial and the approaches reported in the literature only allow considering binary choice situations. We propose an alternative method that can be used in any choice situation. This approach exhibits high flexibility and allows specifying the likelihood function independent of the number of alternatives in the choice-set and in the subset of indifference alternatives. Finally, we test our method with help of four study cases. In the first two experiments (based on simulated and real data), we considered binary choices, observing that the proposed framework allows recovering the parameters used in the generation of the dataset without major difficulties outperforming other alternatives. In case studies 3 and 4 (also on the basis of simulated and real data) we considered further alternatives (specifically opting out). We observe that in most cases the proposed approach allows recovering the real underlying parameters (some problems may arise when considering large indifference thresholds and many indifference observations, but even in this case the proposed approach outperforms competing alternatives). Nevertheless, ignoring the extra variability associated with selecting an alternative as a part of an indifference set does not appear to have a significant effect on the estimated parameters, as is clearly shown in the third and fourth experiments. This way, it may be advisable for modellers to omit this feature, if the data has a scarce number of indifference observations. Further research should provide more evidence regarding this hypothesis. Alternative approaches, such as considering the indifference as an opt-out alternative or just ignoring the indifference preferences are clearly outperformed by the proposed framework and are not capable of recovering the real parameters. Nevertheless, datasets including a very small number of indifference observations do not appear to be severely affected by a poor treatment of indifference. In fact, completely ignoring these observations in case study four does not lead to significantly different results. Hence, the impact of indifference observations depends on their number. Finally, it can be concluded that individuals not stating a clear preference are indeed providing relevant information, which should be considered by the modellers: if you choose not to decide, you still have made a choice. Acknowledgments We wish to thank Ricardo Hurtubia for his useful suggestions and for pointing out the convenience of changing the name of the paper, suggesting we should choose a path that's clear, we should choose freewill. We are also grateful to Juan Pablo Sepúlveda for having provided us with the real data used in our first experiment. Finally, we are indebted to the Institute on Complex Engineering Systems (ICM: P-05-004-F; CONICYT: FB0816), the Centre for Sustainable Urban Development, CEDEUS (Conicyt/Fondap/ 15110020) and the Bus Rapid Transit Centre of Excellence funded by VREF (www.brt.cl), for their support. This article benefited greatly from the helpful comments of two anonymous referees.
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Appendix A See Fig. A1. In this case, respondents were instructed to choose “Any of them” if no alternative was favoured over the other.
Fig. A1. Presentation of choice situations in Case Study 2.
Appendix B See Fig. B1.
Fig. B1. Presentation of choice situations in Case Study 4.
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References Bradley, M.A., Daly, A.J., 1994. Use of the logit scaling approach to test for rank-order and fatigue effects in stated preference data. Transportation 21, 167–184. Bierlaire, M. 2003. BIOGEME: A free package for the estimation of discrete choice models, Proceedings of the 3rd Swiss Transportation Research Conference, Ascona, Switzerland Cantillo, V., Ortúzar, J. de D., 2006. Implications of thresholds in discrete choice modelling. Transp. Rev. 26, 667–691. Cantillo, V., Amaya, J., Ortúzar, J. de D., 2010. Thresholds and indifference in stated choice surveys. Transp. Res. Part B: Methodol. 44, 753–763. Carson, R.T., Louviere, J.J., Anderson, D.A., Arabie, P., Bunch, D.S., Hensher, D.A., Johnson, R.M., Kuhfeld, W.F., Steinberg, D., Swait, J., Timmermans, H., Wiley, J.B., 1994. Experimental analysis of choice. Mark. Lett. 5, 351–367. Chapman, R.G., Staelin, R., 1982. Exploiting rank ordered choice set data within the stochastic utility model. J. Mark. Res. 19, 288–301. Chorus, C., 2012a. Random regret minimization: an overview of model properties and empirical evidence. Transp. Rev. 32, 75–92. Chorus, C.G., 2012b. Logsums for utility-maximizers and regret-minimizers, and their relation with desirability and satisfaction. Transp. Res. Part A: Policy Pract. 46, 1003–1012. Coombs, C.H., Dawes, R.M., Tversky, A., 1970. Mathematical Psychology: An Elementary Introduction. Prentice-Hall, Englewood Cliffs, New Jersey. Dhar, R., 1997. Consumer preference for a no-choice option. J. Consum. Res. 24, 215–231. Dhar, R., Simonson, I., 2003. The effect of forced choice on choice. J. Mark. Res. 40, 146–160. Fenichel, E.P., Lupi, F., Hoehn, J.P., Kaplowitz, M.D., 2009. Split-sample tests of “no opinion” responses in an attribute-based choice model. Land Econ. 85, 348–362. Hess, S., Beck, M.J., Chorus, C.G., 2014. Contrasts between utility maximisation and regret minimisation in the presence of opt out alternatives. Transp. Res. Part A: Policy Pract. 66, 1–12. Kontoleon, A., Yabe, M., 2003. Assessing the impacts of alternative ‘opt-out’ formats in choice experiment studies: consumer preferences for genetically modified content and production information in food. J. Agric. Policy Resour. 5, 1–43. Krishnan, K.S., 1977. Incorporating thresholds of indifference in probabilistic choice models. Manag. Sci. 23, 1224–1233. Louviere, J.J., Hensher, D.A., Swait, J.D., 2000. Stated Choice Methods: Analysis and Applications. Cambridge University Press, Cambridge. McFadden, D., 1974. Conditional logit analysis of qualitative choice behaviour. In: Zarembka, P. (Ed.), Frontiers in Econometrics. Academic Press, New York, 105–142. Olsen, G.D., Swait, J.D., 1998. Nothing is Important. Working Paper, University of Calgary. Quandt, R.E., 1956. A probabilistic theory of consumer behaviour. Q. J. Econ. 70, 507–536. Roberts, J.H., Lattin, J.M., 1991. Development and testing of a model of consideration set composition. J. Mark. Res. 28, 429–440. Schlereth, C., Skiera, B., 2016. Two new features in discrete choice experiments to improve willingness-to-pay estimation that result in SDR and SADR: separated (adaptive) dual response. Manag. Sci.. http://dx.doi.org/10.1287/mnsc.2015.2367. Swait, J.D., Erdem, T., 2007. Brand effects on choice and choice set formation under uncertainty. Mark. Sci. 26, 679–697. Thurstone, L.L., 1927. A law of comparative judgment. Psychol. Rev. 34, 273–286.
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