# An empirical comparison of travel choice models that capture preferences for compromise alternatives

## Abstract

Compromise alternatives have an intermediate performance on each or most attributes rather than having a poor performance on some attributes and a strong performance on others. The relative popularity of compromise alternatives among decision-makers has been convincingly established in a wide range of decision contexts, while being largely ignored in travel behavior research. We discuss three (travel) choice models that capture a potential preference for compromise alternatives. One approach, which is introduced in this paper, involves the construction of a so-called compromise variable which indicates to what extent (i.e., on how many attributes) a given alternative is a compromise alternative in its choice set. Another approach consists of the recently introduced random regret-model form, where the popularity of compromise alternatives emerges endogenously from the regret minimization-based decision rule. A third approach consists of the contextual concavity model, which is known for favoring compromise alternatives by means of a locally concave utility function. Estimation results on a stated route choice dataset show that, in terms of model fit and predictive ability, the contextual concavity and random regret models appear to perform better than the model that contains an added compromise variable.

### Keywords

Compromise alternatives Route-choices Random regret Contextual concavity## Introduction

Compromise alternatives have an intermediate performance on each or most attributes rather than having a poor performance on some attributes and a strong performance on others (relative to other alternatives in the choice set). In fields adjacent to transportation, most notably in consumer research,^{1} the preference among decision makers for compromise alternatives has been well established empirically as one of the most important and persistent choice set-composition effects (e.g., Simonson 1989; Wernerfelt 1995; Kivetz et al. 2004; Müller et al. 2010). Attempts to capture this behavioral phenomenon have led to the development of different choice models that allow for capturing the popularity of compromise alternatives (e.g., Kivetz et al. 2004); the arguably most elegant and effective of these models is the so-called Contextual Concavity model or from here on CCM (Kivetz et al. 2004). Although it seems quite obvious that also in many transportation contexts compromise alternatives exist, virtually no attention has been paid so far in our field to the development and testing of choice models that allow for capturing their relative popularity in the context of transportation related decision making.

This paper discusses and empirically tests the above mentioned CCM and two alternative discrete (travel) choice models that allow for capturing the relative popularity of compromise alternatives. One approach is based on creating a so-called compromise-variable for each alternative, which indicates to what extent (i.e., on how many attributes) an alternative is a compromise alternative in a given choice set. This approach is new, to the best of the authors’ knowledge.^{2} A second approach is based on the recently introduced Random Regret Minimization-approach or RRM from here on; see Chorus (2010) and Chorus (2012a). Although theoretical derivations and numerical examples presented in the two cited papers have suggested that Random Regret-based travel choice models are theoretically expected to assign relatively high choice probabilities to compromise alternatives, these papers do not present an empirical analysis of that claim. Chorus and Rose (2011) do so, as they highlight how an estimated RRM-model favors compromise alternatives in the context of online date-selection data. However, in contrast with this paper, Chorus and Rose (2011) do not present out-of-sample tests of predictive performance, nor do they compare the RRM-model with alternative model forms other than the linear-in-parameters logit model.

It should be stated up front that this paper is not concerned with empirically exploring the potential presence of what has been called a ‘compromise effect’ in our data. The compromise effect (Simonson 1989) states that the ratio of an alternative’s choice probability and another alternative’s choice probability is larger in a choice set where the former is a compromise alternative, relative to a choice set where the latter is a compromise alternative.^{3} Strictly speaking, our data do not allow for finding unambiguous evidence of the presence or absence of compromise effects, as the dataset did not feature the appearance of a particular alternative (bundle of attribute values) as a compromise alternative in one choice task, and as an extreme alternative in another task. However, as will be shown below our data do allow for testing the empirical performance of models that allow for capturing the (potential) popularity of compromise alternatives.

Section 2 introduces the CCM-model and the two alternative modeling approaches described above as alternatives for the CCM-approach. In Sect. 3 and 4 the mentioned modeling approaches are compared (also with a linear-in-parameters logit model that does not allow for capturing any relative popularity of compromise alternatives) in the context of stated route choice data collected recently among a sample of Dutch commuters. Subsequently, estimation results Sect. (3) and out-of-sample validation results Sect. (4) are presented and discussed in detail. Conclusions and discussion are provided in Sect. 5.

## Three models that capture potential preferences for compromise alternatives

### A logit-model with a compromise variable

*J*alternatives each described in terms of

*M*attributes, the compromise variable \( C_{i} \) can be denoted as follows (where

*m*stands for an attribute,

*x*represents a level, and

*i*an alternative):

^{4}

### A regret-based logit-model

The second approach used in this paper to accommodate preferences for compromise alternatives is based on the recently developed Random Regret Minimization-approach (RRM) to discrete choice modeling (Chorus 2010). The RRM-model postulates that, when choosing between alternatives, decision-makers aim to minimize anticipated random regret, and that the level of anticipated random regret that is associated with the considered alternative *i* is composed of an i.i.d. random error, which represents unobserved heterogeneity in regret and whose negative is Extreme Value Type I-distributed, and a systematic regret. Systematic regret is in turn conceived to be the sum of all so-called binary regrets that are associated with bilaterally comparing the considered alternative with each of the other alternatives in the choice set.

*j*is conceived to be the sum of the regrets that are associated with comparing the two alternatives in terms of each of their

*M*attributes. This attribute level-regret in turn is formulated as \( \ln \left( {1 + { \exp }\left[ {\beta_{m} \cdot \left( {x_{jm} - x_{im} } \right)} \right]} \right) \). This formulation implies that regret is close to zero when alternative

*j*performs (much) worse than

*i*in terms of attribute

*m*, and that it grows as an approximately linear function of the difference in attribute-values in case

*i*performs worse than

*j*in terms of attribute

*m*. In that case, the estimable parameter

*β*

_{m}(for which also the sign is estimated) gives the approximation of the slope of the regret-function for attribute

*m*. Systematic regret is thus written as:

Acknowledging that minimization of random regret is mathematically equivalent to maximizing the negative of random regret, choice probabilities may be derived using a variant of the logit formulation: the choice probability associated with *i* equals \( P\left( i \right) = \frac{{{ \exp }( - R_{i} )}}{{\mathop \sum \nolimits_{j = 1..J} { \exp }( - R_{j} )}} \). Note that the resulting model consumes as many parameters as a conventional linear-in-parameters logit-model, and that the model can be estimated using standard software-packages such as Biogeme and NLOGIT (version 5). The RRM-model has been tested empirically in a wide range of decision-contexts including but not limited to mode-, route-, departure time-, vehicle type-, and destination-choices. See Chorus (2012a) for a concise overview of this empirical evidence.^{5}

One important difference between the RRM-model and the conventional linear-in-parameters logit model is that the RRM-model implies a particular type of semi-compensatory behavior. This is a direct result of the convexity of the regret-function: improving an alternative in terms of an attribute on which it already performs well relative to other alternatives generates only small decreases in regret. Deteriorating to a similar extent the performance on another equally important attribute on which the alternative has a poor relative performance may generate substantial increases in regret. As a result, the extent to which a strong performance on one attribute can make up for a poor performance on another depends on the relative position of each alternative in the set.

As explained in Chorus (2010) and (2012a), RRM’s theoretical ability to accommodate a preference for compromise alternatives follows directly from this particular type of semi-compensatory behavior. More specifically, the RRM-model predicts that having a (very) poor performance on one attribute causes much regret, while having a (very) strong performance on another attribute does not necessarily compensate for this poor performance. As a result, RRM-models predict that it is more efficient (in terms of avoiding regret and gaining market share) to be a compromise alternative: even when a compromise alternative fails to have a strong performance on any of the attributes (relative to the other alternatives in the set), RRM-models predict that it will still only generate modest levels of regret as long as it does not have a particularly poor performance on any of the attributes.

The RRM model has in common with the CCM (see below) that ‘non-compromise’ alternatives are treated asymmetrically: it pays off more to avoid having a relatively poor performance on any attribute, than it does to achieve a very strong performance on one or more attributes. In contrast with the CCM, the RRM model generates preferences for compromise alternatives by definition, and irrespective of estimated parameter values. Whereas the CCM is capable of capturing preferences for compromise alternatives, extreme alternatives, or none of these (depending on the estimated value for the concavity parameter), the RRM model structure postulates a preference for compromise alternatives. To avoid repetition, and for reasons of space limitations, we refer to Chorus (2010) and (2012a) for a more elaborate and formal discussion of how the RRM-model accommodates preferences for compromise alternatives.

### The contextual concavity model

*J*alternatives each described in terms of

*M*attributes; index

*i*represents an alternative, index

*m*indicates an attribute):

Here, \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{x}_{m} \) stands for the least preferred level of attribute *m* in the context of the given choice task (in practice this is the attribute value which the analyst *believes* to be the least preferred), and \( \varphi^{m} \) is an attribute-specific concavity parameter which is expected (but not constrained) to be smaller than 1 and positive. As highlighted by Kivetz et al. (2004), the CCM-model with a concavity parameter between 0 and 1 assigns a relatively high choice probability to compromise alternatives by means of downwardly adjusting the utilities of the best performing attributes to a greater extent than the utilities of attributes with an intermediate performance. This is due to the concavity of the utility function. As such, CCM’s treatment of ‘non-compromise alternatives’ is intrinsically asymmetric: when \( \varphi^{m} \in \left] {0,1} \right[ \), it is more beneficial (in terms of attaining a higher choice probability) to avoid a relatively poor performance on any attribute, than it is beneficial to achieve a very strong performance on one or more attributes.

As Kivetz et al. (2004) note, when \( \varphi^{m} \) equals 1, the CCM model is equivalent to a linear-additive logit model and when \( \varphi^{m} \) is greater than 1, there is a preference for extreme alternatives rather than compromise alternatives. Note that the term ‘contextual’ refers to the fact that the concavity is exhibited relative to the least preferred attribute level. By adding i.i.d. errors (Extreme Value Type I distributed) to the observed utilities, a logit-model is obtained. To avoid repetition, and for reasons of space limitations, we refer to Kivetz et al. (2004) for a more elaborate and formal discussion of how the CCM assigns higher choice probabilities to compromise alternatives.

Before moving to the empirical part of the paper, it should be noted that there is a fundamental difference between the CCM- and RRM-models on the one hand, and the compromise variable-model on the other hand. That is, in the first two models preferences for compromise alternatives emerge from behavioral premises that themselves do not directly or explicitly refer to compromise alternatives or compromise seeking behavior. In the CCM-model these preferences emerge from locally concave utility functions, and in the RRM-model they emerge from convex regret-functions. In contrast, the compromise variable-model postulates explicitly that individuals are focused on identifying and choosing compromise alternatives. It may also be noted at this point that of the three models that generate preferences for compromise alternatives, the RRM is the most parsimonious: it consumes no more parameters than a linear-in-parameters logit model.

## A stated route-choice experiment

The data collection effort focused on route choice behavior among commuters who travel from home to work by car. A total of 550 people were sampled from an internet panel maintained by IntoMart, in April 2011. Sampled individuals were at least 18 years old, owned a car, and were employed. It was taken care of that the sample was representative for the Dutch commuter in terms of gender, age and education level. Of these 550 people, 390 filled out the survey (implying a response rate of 71 %).

Respondents to the survey were asked to imagine the hypothetical situation where they were planning a new commute from home to work (either because they had recently moved, or because their employer had recently moved, or because they had started a new job). They were asked to choose between three different routes that differed in terms of the following four attributes, with three levels each: average door-to-door travel time (45, 60, 75 min), percentage of travel time spent in traffic jams (10, 25, 40 %), travel time variability (5,15, ± 25 min), and total costs (5.5, 9, €12.5). Note that for these attributes the least-preferred values per choice task are easily identified, since it may be safely assumed that for each attribute, lower values are preferred over higher ones.

An example route choice-task featuring a full-fledged compromise alternative

1 | Route A | Route B | Route C |
---|---|---|---|

Average travel time (minutes) | 45 | 60 | 75 |

Percentage of travel time in congestion (%) | 10 % | 25 % | 40 % |

Travel time variability (minutes) | ±5 | ±15 | ±25 |

Travel costs (Euros) | €12,5 | €9 | €5,5 |

YOUR CHOICE | □ | □ | □ |

An example route choice-task without a full-fledged compromise alternative

7 | Route A | Route B | Route C |
---|---|---|---|

Average travel time (minutes) | 60 | 75 | 45 |

Percentage of travel time in congestion (%) | 10 % | 25 % | 40 % |

Travel time variability (minutes) | ±15 | ±25 | ±5 |

Travel costs (Euros) | €5,5 | €12,5 | €9 |

YOUR CHOICE | □ | □ | □ |

In this latter choice set, alternative A is still a compromise-alternative, but only to a limited extent: it has an extreme performance on two out of four attributes (congested travel time and travel costs).

## Empirical analyses: model estimation and validation

### Model estimation

Four models were estimated on the choice data, using the Biogeme-freeware package (Bierlaire 2003, 2008): first, a linear-in-parameters logit-model was estimated (Model I). Then, the compromise variable was added to the equation (Model II). Subsequently, the RRM-model was estimated (Model III). Finally, the CCM-model is estimated (Model IV).

With respect to Model II, it may be noted that we also estimated a model where the parameter associated with the compromise variable was allowed to vary randomly between respondents (but not within choices made by the same respondent). This panel-specification allows for compromise preferences to be considered as personality traits. The panel model resulted–as is to be expected–in a slightly higher model fit and a significant (though not large) estimated value for the standard deviation of the compromise parameter. Other estimation results remain similar to those obtained for the non-panel model. To allow for a meaningful comparison across the four models, we have chosen not to consider the panel-specification in the remainder of this paper, and focus on the logit specifications instead.

With respect to Model III and Model IV, it may be noted that we also tested for significant compromise parameters. However, in the context of the CCM-model the associated parameter was insignificant at any reasonable level. In the context of the RRM-model, the associated parameter was only significant at a 10 %-level. Furthermore, the RRM-model with the added compromise variable performed worse than the RRM-model without compromise variable in terms of every singly test of out-of-sample predictive ability. Therefore we only report results for the CCM-model and RRM-model without compromise variable.

Estimation results (robust t-values between brackets)

Model I base-model | Model II base-model + compromise-variable | Model III RRM-model | Model IV CCM-model | |
---|---|---|---|---|

Average travel time | −0.0673 (−35.13) | −0.0695 (−34.21) | −0.0468 (−32.50) | −0.0697 (−33.46) |

% of travel time in congestion | −0.0273 (−17.39) | −0.0295 (−17.42) | −0.0181 (−16.66) | −0.0358 (−11.83) |

Travel time variability | −0.0316 (−11.86) | −0.0320 (−11.92) | −0.0210 (−11.86) | −0.0314 (−11.91) |

Travel costs | −0.173 (−21.52) | −0.164 (−19.36) | −0.113 (−20.28) | −0.241 (−5.78) |

Compromise-variable | – | 0.1100 (5.27) | – | – |

Concavity par. (travel costs) | – | – | – | 0.6984 (5.47) |

Concavity par. (% time spent in congestion) | – | – | – | 0.422 (5.57) |

Nr of cases | 3,510 | 3,510 | 3,510 | 3,510 |

Final-LL | −2,613 | −2,600 | −2,605 | −2,589 |

Adj. Rho-square | 0.321 | 0.324 | 0.323 | 0.327 |

^{6}) mean elasticities which give the percentage change in choice probability resulting from a percentage change in one of the alternative’s attribute’s levels (Tables 4 and 5): also here, differences between models with- and without the compromise variable are small.

Direct elasticities (95 %-confidence intervals between brackets)

Model I base-model | Model II base-model + compromise-variable | Model III RRM-model | |
---|---|---|---|

Average travel time | −2.82 [−3.40, −2.28] | −2.91 [−4.42, −1.57] | −3.03 [−3.65, −2.44] |

% of travel time in congestion | −0.47 [−0.56, −0.40] | −0.51 [−0.75, −0.26] | −0.48 [−0.57, −0.41] |

Travel time variability | −0.32 [−0.36, −0.29] | −0.33 [−0.41, −0.26] | −0.33 [−0.37, −0.29] |

Travel costs | −1.06 [−1.12, −1.00] | −1.00 [−1.13, −0.88] | −1.06 [−1.11, −1.01] |

Cross elasticities (95 %-confidence intervals between brackets)

Model I base-model | Model II base-model + compromise-variable | Model III RRM-model | |
---|---|---|---|

Average travel time | 1.12 [0.92, 1.34] | 1.16 [0.64, 1.74] | 1.19 [1.00, 1.38] |

% of travel time in congestion | 0.18 [0.15, 0.22] | 0.20 [0.11, 0.30] | 0.19 [0.16, 0.22] |

Travel time variability | 0.14 [0.12, 0.16] | 0.14 [0.10, 0.19] | 0.14 [0.12, 0.17] |

Travel costs | 0.45 [0.40, 0.50] | 0.43 [0.31, 0.55] | 0.46 [0.41, 0.51] |

The reason for including RRM-elasticities in Tables 4 and 5 is that, in contrast with parameter values themselves which cannot be compared directly between linear-in-parameters logit-models and RRM-models,^{7} elasticities can be compared across regret- and utility-based model types. Tables 4 and 5 show that elasticities are roughly equal across the RRM- and linear-in-parameters logit models (Model I and Model III). This is in line with findings from previous studies (Thiene et al. 2012; Hensher et al. in press) although these latter two studies reported differences that were somewhat larger than those we found on our data.

In terms of model fit, it is seen that differences between models are small. More specifically, it appears that in the context of our data the RRM-model (Model III) outperforms the linear-in-parameters logit-model (Model I), while a still better model fit is obtained by Model II (linear-in-parameters logit-model + compromise variable). Model IV (CCM-model) achieves the highest fit, also when correcting for the number of parameters. Using the Likelihood-ratio statistic for nested models (Models I, II, and IV) and Ben-Akiva and Swait’s (1986) test for nonnested models (model III versus the other three models) it is found that all differences in model fit are significant at a 5 %-level.

Choice probability predictions (percentages, rounded; based on routes presented in Table 1; 95 % confidence intervals between brackets)

Route A | Route B | Route C | |
---|---|---|---|

Model I | 70 | 23 | 7 |

Base-model | [65, 76] | [19, 25] | [5, 10] |

Model II | 64 | 30 | 5 |

Base-model + compromise-variable | [49, 77] | [20, 38] | [2, 13] |

Model III | 67 | 27 | 6 |

RRM-model | [62, 72] | [24, 29] | [4, 9] |

Model IV | 66 | 66 | 5 |

CCM-model |

To test whether these results are statistically significant, we also computed 95 %-confidence intervals. This was done by making 100 draws from the multivariate normal distribution of the estimates (as implied by their means, standard errors and covariances) and computing choice probabilities for routes A, B and C for each multi-dimensional draw. The interval spans the values from the 5 and the 95 % quantiles of the generated values.^{8} Note that no confidence intervals for the choice probabilities of the CCM could be obtained, as the simulation effort suffered from stability issues in the context of the that model. The resulting confidence intervals suggest that the Models I and III (the base model and the RRM-model) provide significantly different forecasts for the choice probability of the compromise alternative (route B), as Model I’s mean prediction for B is not in Model III’s 95 % confidence interval and vice versa. Differences between models I and II are not significant.

Another result can be obtained from the confidence intervals: it appears that Model III (the RRM model) is more precise than Model II (linear-in-parameters logit with compromise variable) and has a similar precision as Model I (the base model). This suggests an advantage, in terms of precision, of the RRM model.

### Model validation: out-of-sample predictions

Predictive ability out-of-sample (best results per metric underlined)

Model I base-model | Model II base-model + compromise-variable | Model III RRM-model | Model IV CCM-model | |
---|---|---|---|---|

Mean likelihood of chosen alternative | 0.555 | 0.555 | 0.555 | 0.554 |

Log-likelihood of validation sample | −913 | −910 | −913 | −905 |

Rho bar squared | 0.300 | 0.301 | 0.300 | 0.304 |

Ability to predict market shares of compromise alternatives (best results per metric underlined)

Model I base-model | Model II base-model + compromise-variable | Model III RRM-model | Model IV CCM-model | |
---|---|---|---|---|

Mean deviation (percentage points) | 1.7 | −1.3 | 0.0 | −0.7 |

Mean absolute deviation (percentage points) | 4.2 | 3.3 | 3.0 | 3.1 |

Root mean squared error (RMSE) (percentage points) | 4.8 | 4.1 | 3.8 | 3.6 |

Starting with the mean deviation, it appears that the linear-in-parameters logit model without a compromise variable underestimates the choice probability of compromise alternatives, while the addition of a compromise variable leads to an overestimation of compromise alternatives’ choice probabilities. This finding is in line with intuition. The RRM-model appears to very accurately predict the market share of compromise alternatives when evaluated in terms of this metric. Also when considering the mean absolute deviation, the RRM-model appears to outperform the other models, although the difference with the CCM-model is very small. In terms of the RMSE, the CCM-model appears to do best, closely followed by the RRM-model.

^{9}For these alternatives predicted choice probabilities were compared to relative choice frequencies. Table 9 presents results (note that as far as the mean-deviation criterion is concerned, we expect opposite signs compared to those in Table 8).

Ability to predict market shares of ‘extreme’ alternatives (best results per metric underlined)

Model I base-model | Model II base-model + compromise-variable | Model III RRM-model | Model IV CCM-model | |
---|---|---|---|---|

Mean deviation (percentage points) | −1.2 | 1.7 | 0.1 | 0.5 |

Mean absolute deviation (percentage points) | 3.7 | 3.2 | 2.9 | 3.0 |

Root mean squared error (RMSE) (percentage points) | 4.1 | 4.1 | 3.7 | 3.6 |

It is easily seen that results reported in Table 9 closely resemble those reported in Table 8, at least as far as orderings in terms of model performance are concerned. This suggests that, at least in the context of our data, the ability of the CCM and RRM model in terms of accommodating preferences for compromise alternatives does not appear to come at the cost of biased predictions regarding choice probabilities for more extreme alternatives.

## Conclusions and discussion

This paper presents and tests models that are able to capture potential preferences for so-called compromise alternatives, in the context of route choice data. (compromise alternatives have an intermediate performance on each or most attributes rather than having a poor performance on some attributes and a strong performance on others.) One approach involves the construction of a so-called compromise variable, which indicates to what extent (i.e., on how many attributes) a given alternative is a compromise alternative in its choice set. Another approach consists of using a Random Regret-model form (RRM). In the context of this latter model-type, preferences for compromise alternatives emerge from the underlying model structure. The two proposed model forms are compared with the Contextual Concavity-model (CCM), which has shown its worth in terms of generating preferences for compromise alternatives by means of a locally concave utility function. The comparison is based on a stated route choice dataset.

In terms of model fit, the CMM appears to have a slight (but statistically significant) edge over the other presented model forms, also when correcting for the number of parameters which is higher for the CMM-model. A validation exercise on one third of the data that was not used for estimation shows that differences between models in terms of generic out-of-sample predictive ability are small; a mixed picture emerges regarding the performance of the different models, although the CCM and the RRM-model appear to outperform the other models (among these two models, no clear ‘winner’ can be identified).

In this light it is worth noting that the RRM-model is the most parsimonious model of the four models used (together with the relatively poor performing linear-in-parameters logit-model). This suggests that the RRM-model may be considered a viable alternative for the CCM-model, especially when the number of attributes increases (because increases in attribute-numbers imply increases in the number of parameters used by the CCM-model if attribute-specific concavity parameters are estimated). As another more general reflection, our findings suggest that the two models where preferences for compromise alternatives emerge indirectly from model structures and behavioral premises (such as locally concave utility functions or convex regret functions) perform better empirically than the model that explicitly postulates (by means of a compromise-variable) that decision-makers identify and choose compromise alternatives.

A number of directions for future research come to mind. First, there is an obvious need to study to what extent results obtained in this study hold in the context of other travel choice-related datasets (stated and revealed). Moreover, it would be interesting to find out if there are circumstances or personality traits that trigger the kind of decision-making that results in preferences for compromise alternatives. In this light, it is interesting to point at the fact that the stated choice experiment reported in this paper also included a number of Likert-scale questions to respondents concerning their attitude to decision-making in general and in the context of the stated choice experiment. We found that on the sub-sample of individuals that (stated that they) found it important to make the ‘right’ choices in the experiment, a large and significant parameter associated with the compromise variable was found, while this parameter was insignificant in the context of the sub-sample of individuals that (stated that they) did not find it important to make the ‘right’ choices in the experiment. Similarly, we found that on the sub-sample of individuals that (stated that they) found it difficult to make decisions when the stakes are high, a large and significant parameter associated with the compromise variable was found, while the parameter was insignificant on the sub-sample of individuals that (stated that they) did not find it difficult to make decisions when the stakes are high.

Although in the context of other similar Likert-scale questions no significant differences between sub-samples were obtained, this does suggest that the ability of models to generate preferences for compromise alternatives might be particularly important in the context of choices that are considered by decision-makers to be important and/or difficult. Note that these results are in line with findings reported in the consumer choice literature (e.g., Simonson 1989). More research is needed to test these preliminary findings.

## Footnotes

- 1.
More specifically, preferences for compromise alternatives have been shown in the context of choices among apartments, among mouthwashes, among political manifestos, among investment portfolios, among on-line dates and among laptops, to name just a few examples.

- 2.
Note that Simonson (1989) presented a model that adds a dummy-variable to an alternative’s utility function. The dummy-variable equaled 1 if an alternative was a compromise alternative in terms of both attributes considered in that study, and 0 otherwise. This approach resembles our approach, but is less sensitive, especially when the number of attributes increases.

- 3.
More formally: consider two choice sets containing three alternatives each:

*I*= {*A*,*B*,*C*} and*II*= {*B*,*C*,*D*}. Assume that each of the alternatives is evaluated in terms of two observable attributes*x*_{1}and*x*_{2}(and assume for ease of communication that higher attribute values are preferred over lower ones). Assume furthermore that*x*_{1}^{A}<*x*_{1}^{B}<*x*_{1}^{C}<*x*_{1}^{D}and that*x*_{2}^{A}>*x*_{2}^{B}>*x*_{2}^{C}>*x*_{2}^{D}. According to the conceptualizations in Simonson (1989) and Kivetz et al. (2004), alternative*B*is a compromise alternative in choice set*I*since*x*_{1}^{A}<*x*_{1}^{B}<*x*_{1}^{C}and*x*_{2}^{A}>*x*_{2}^{B}>*x*_{2}^{C}. In set*II*,*C*is a compromise alternative since*x*_{1}^{B}<*x*_{1}^{C}<*x*_{1}^{D}and*x*_{2}^{B}>*x*_{2}^{C}>*x*_{2}^{D}. Then, the compromise effect as formulated by, for example, Simonson (1989) and Kivetz et al. (2004) states that*P*_{B}^{I}/*P*_{C}^{I}>*P*_{B}^{II}/*P*_{C}^{II}, where*P*_{B}^{I}denotes the choice probability associated with alternative*B*in the context of choice set*I*, etc. - 4.
Note that this model differs from both the CCM and RRM models introduced below as it treats ‘non-compromise’ alternatives symmetrically. That is, the model does not distinguish between attribute-scores that are either lower or higher than the compromise score (besides of course through conventional parameters

*β*_{m}). It could be possible to induce asymmetry in this model by means of constructing variables that keep track of the number of attributes on which an alternatives scores better (or: worse) than the compromise values for the respective attributes. However, it turns out, not entirely unexpected, that these asymmetric effects are hard to identify (at least in the context of our data), as the estimated asymmetry-parameters appear to be strongly correlated with parameters measuring attribute importance in the conventional way (i.e., parameters*β*_{m}). As a result, this asymmetric model form is not considered in the remainder of this paper, although testing this asymmetric model variant on other datasets may be considered a fruitful direction for further research. - 5.
Importantly, in contrast with other (travel) models and theories that are based on regret-minimization such as for example the ones presented in Chorus (2012b) and Fonzone et al. (2012), the RRM-model focuses on so-called riskless choices (that is, choice situations in which the decision maker is assumed to know with certainty the values of the attributes of alternatives). The RRM model postulates that as long as alternatives are characterized in terms of multiple attributes, this implies that trade-offs have to be made by the decision-maker, and that as a result there will be regret in the sense that there will generally be at least one non-chosen alternative that outperforms a chosen one in terms of one or more attributes. Note though, that the RRM model can easily be extended to cover risky decision making as well, as explained in Chorus (2012a).

- 6.
Since the alternatives in the experiment were unlabeled, only cross-elasticities with respect to one combination of alternatives (more specifically: A versus B) is considered.

- 7.
In a (linear-in-parameters) RUM formulation, a parameter represents the amount of utility that is gained or lost by increasing or decreasing the performance in terms of the corresponding attribute by one unit. Quite differently, in a RRM-setting parameters reflect the

*upper bound*of the extent to which a unit in- or decrease in performance on an attribute influences (bilateral) regret. Whether or not this upper bound is reached for a one unit in- or decrease in an attribute’s value depends on the performance of other alternatives in the set in terms of the attribute. - 8.
- 9.
Note that if the two ‘non-compromise’ alternatives contained an equal number of attributes with an extreme score, their summed relative choice frequency was compared to the summed predicted choice probabilities.

## Notes

### Acknowledgments

Support from The Netherlands Organization for Scientific Research (NWO), in the form of VENI-grant 451.10.001, is gratefully acknowledged by the first author. Comments made by four anonymous reviewers have helped us improve a previous version of this paper.

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