Abstract
This Chapter presents the RRM-model. First, the Random Regret-function is presented and explained (Sect. 2.1). Subsequently, this function is compared with the classical linear-additive Random Utility-function (Sect. 2.2). Finally, it is shown how the Random Regret-function translates into MNL-type choice probabilities for a particular distribution of the random error terms (Sect. 2.3).
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Notes
- 1.
Obviously, the level of anticipated regret that is associated with a particular alternative will vary between individuals. More specifically, different individuals may have different tastes and perceptions regarding alternatives and their attributes. Mathematically, this heterogeneity across individuals can be expressed by making relevant terms in the regret equation presented below (such as β and ε) individual-specific by means of an index (usually ‘n’). In this tutorial, for reasons of readability, no such indices are used. As a result, equations refer to (the tastes and perceptions of) an average or ‘representative’ individual.
- 2.
It should be noted at this point that during the estimation process the sign of parameters is estimated together with their magnitude. That is, no a priori expectations need to be formulated by the analyst in terms of whether higher attribute-values are preferred by the decision-maker over lower ones, or vice versa.
- 3.
- 4.
Note that, although the use of smooth attribute-regret-function instead of the non-smooth one is inspired mostly by pragmatic reasons as argued above, there is a deeper connection between the two functions as well. That is, when ignoring a constant, ln(1 + exp[β m · (x jm − x im )]) gives the expectation of max{0, [β m · (x jm − x im )]} when the two terms between curly brackets are considered i.i.d. random variables with Extreme Value Type I-distribution (having a variance of π2/6). A reason for this stochasticity might be that the researcher is only able to assess these two terms up to a random error.
- 5.
Note that strictly speaking, DCMs do not really assume particular processes (in the sense that they do not postulate a particular order of decision-making steps). Rather, the mathematical formulation of the linear-additive RUM- model is in fact consistent with a range of underlying decision processes. Nonetheless, throughout the literature the linear-additive RUM-model form is generally considered to be the mathematical representation of the decision process described and visualized on this and the next page. It is instructive at this point to assume this particular order in decision-making steps as it highlights the ways in which RRM- and RUM-based decision rules differ in a conceptual sense.
- 6.
Note that the random error is ignored in this example as it refers to the analyst’s lack of knowledge and as such is irrelevant from an individual decision-maker’s point of view.
- 7.
The term i.i.d. stands for identically and independently distributed. This means that errors assigned to different alternatives are uncorrelated, and are drawn from the same distribution (with the same variance). This variance is usually fixed to π2/6, which indirectly implies a normalization of systematic utility. In this tutorial, the scale of the utility or regret is always normalized this way, and is therefore not explicitly mentioned in equations.
- 8.
See the Appendix for a discussion of the validity of the assumption of i.i.d. errors in the context of RRM-models.
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Chorus, C.G. (2012). A Random Regret Minimization-based Discrete Choice Model. In: Random Regret-based Discrete Choice Modeling. SpringerBriefs in Business. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29151-7_2
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