Appendix: Theory and Methodology

Appendix: Theory and methodology

Background of regret minimisation

Evidence for regret minimisation has already been provided in many empirical and theoretical studies since several decades [1-4]. Most of these studies infer regret minimisation premises from observed choices, while others have even tried to find evidence for regret based decision making using neuro-imaging techniques. Take for example [5] who show that the area of the human brain that is active when individuals experience regret after having made a (poor) choice, is also highly active split seconds before they make a choice. In their words “anticipating regret is a powerful predictor of future choices”.

Random regret minimisation model

The random regret minimisation (RRM) model is designed to incorporate the following seven behavioural intuitions relating to the anticipated regret associated with a considered alternative [6]:

1.  when a considered alternative outperforms another alternative in terms of a particular attribute there is no anticipated regret.

2.  when a considered alternative is outperformed by another alternative in terms of a particular attribute there is anticipated regret.

3.  anticipated regret increases with the importance of the attribute on which a considered alternative is outperformed by another alternative.

4.  anticipated regret increases with the magnitude of the extent to which a considered alternative is outperformed by another alternative on a particular attribute.

5.  anticipated regret increases with the number of attributes on which the considered alternative is outperformed by another alternative.

6.  anticipated regret increases with the number of alternatives that outperform a considered one on a particular attribute.

7.  anticipated regret is, from the perspective of the analyst, partially ‘observable’ (in the sense that it can be explicitly linked to observed variables) and partially ‘unobservable’.

The following equation gives a formulation of regret that is consist with these intuitions:

The term is the core of equation 3. It forms a measure of the amount of regret that is associated with comparing a considered alternative i with another alternative j in terms of a particular attribute . This formulation implies that regret approaches zero when alternative j performs (much) worse than i in terms of attribute , and that it grows as a linear function of attribute importance and the difference in performance between alternatives on the attribute, in case alternative i performs (much) worse than j in terms of attribute . The estimable parameter gives (the approximation of) the slope of the regret-function for attribute . In other words: when the considered alternative is outperformed by a competing alternative in terms of a particular attribute, attribute-level regret increases as the attribute becomes more important, and as the difference in performance between the attributes becomes larger. This attribute-level regret is computed for each of the bilateral comparisons with other alternatives, and for all available attributes. This so-called Logsum-formulation of attribute regret provides a smooth and close approximation of the following non-smooth measure of attribute-level regret: . Although this latter non-smooth measure is a more intuitive measure of attribute-level regret, the max-operator causes problems when equals zero. More specifically, the fact that the function is discontinuous around zero has been shown to create difficulties with respect to the derivation of marginal effects and with respect to maximum likelihood-based model estimation [7]. When adopting the smooth approximation (i.e., Eq. 3), the RRM approach becomes not only parsimonious, valuable and feasible to model choice data, but it can also be easily coded and estimated using standard discrete-choice software packages (e.g. the software package NLOGIT (version 2012), or Biogeme [8]; see also [6] for several examples of syntaxes). As explained and illustrated in [6] and [7], the approximation is quite close to the max-formulation of regret, especially when regret is either large or close to zero.

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, gives the expectation of 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).

References

1. Bell DE. Regret in decision making under uncertainty. Operations Research. 1982;30(5):961-81.

2. Chorus CG, Molin EJE, van Wee GP, Arentze T.A., Timmermans H.J.P. Responses to transit information among car-drivers: regret-based models and simulations. Transportation Planning and Technology. 2006;29(4):249-71.

3. Loomes G, Sugden R. A rationale for preference reversal. American Economic Review. 1983;73(3):428-32.

4. Simonson I. The influence of anticipating regret and responsibility on purchasing decisions. Journal of Consumer Research. 1992;19(1):105-19.

5. Corricelli G, Critchley HD, Joffily M, O'Doherty JP, Sirigu A, Dolan RJ. Regret and its avoidance: A neuroimaging study of choice behaviour. Nature Neuroscience. 2005; 8(9): 1255-1662

6. Chorus CG. Random regret-based discrete choice modelling: a tutorial.: Springer; 2012.

7. Chorus CG. A new model of Random Regret Minimization. European Journal of Transport and Infrastructure Research. 2010;10(2):181-96.

8. Bierlaire M. BIOGEME: A free package for the estimation of discrete choice models, Proceedings of the 3rd Swiss Transportation Research Conference Ascona, Switzerland; 2003.

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